You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Given $\geq 3$, $1<p<N$, two measurable functions $V\left(r \right)\geq 0$, $K\left(r\right)> 0$, and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation $$-\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}.$$ We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1 -1}, t^{q_2 -1}  \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.'
author:
- 'Marino Badiale[^1] ^,^[^2]  - Michela Guida^a^  - Sergio Rolando[^3]'
date:  
title: Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials
---

Introduction
============

In this paper we pursue the work we made in papers [@BGR_I; @BGR_II; @BGR_p; @BZ; @GR-nls; @BGR_bilap], where we studied embedding and compactness results for weighted Sobolev spaces. These results then made possible to get existence and multiplicity results, by variational methods, for several kinds of elliptic equations in $\mathbb{R}^N$.

In the present paper we face quasilinear elliptic equations in presence of a radial potential on the derivatives, that is, equations of the following kind $$\label{EQ}
-\mathrm{div}\left(A(|x|) |\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}$$ where $1<p<N$, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous nonlinearity satisfying $f\left( 0\right) =0$, and $V, A, K$ are given potentials. We study such equation by variational methods, so we introduce a suitable functional space $X$ (see section 2) and we say that $u\in X$ is a *weak solution *to (\[EQ\]) if $$\int_{\mathbb{R}^{N}}|\nabla u|^{p-2}\nabla u\cdot \nabla h\,dx+\int_{\mathbb{R}^{N}}V\left(
\left| x\right| \right) |u|^{p-2}uh\,dx=\int_{\mathbb{R}^{N}}K\left( \left| x\right|
\right) f\left( u\right) h\,dx\quad \textrm{for all }h\in X.
$$ These solutions are (at least formally) critical points of the Euler functional $$I\left( u\right) :=\frac{1}{p}\left\| u\right\| ^{p}-\int_{\mathbb{R}
^{N}}K\left( \left| x\right| \right) F\left( u\right) dx,  \label{I:=}$$ where $F\left( t\right) :=\int_{0}^{t}f\left( s\right) ds$ and $|| \cdot ||$ is the norm on $X$ (see section 2 below). Then the problem of existence is easily solved if $A\equiv 1$, $V$ does not vanish at infinity, $K$ is bounded and $f(t)= t^{q-1}$, because standard embeddings theorems for $X$ are available (for suitable $q$’s). As we let $V$ and $K$ to vanish, or to go to infinity, as $|x| \rightarrow 0$ or $|x|\rightarrow +\infty$, and we introduce the potential $A$ on the derivatives, the usual embeddings theorems for Sobolev spaces are not available anymore, and new embedding theorems need to be proved. This has been done in several papers: see e.g. the references in [@BGR_I; @BGR_II; @GR-nls] for a bibliography concerning the usual Laplace equation, [@Anoop; @Su12; @Cai-Su-Sun; @SuTian12; @Su-Wang-Will-p; @Yang-Zhang; @Zhang13; @BPR] for equations involving the $p$-laplacian, and [@BZ; @Su-Wang] and the references therein for problems with a potential $A$ on the derivatives.

The main novelty of our approach (in [@BGR_I; @BGR_II; @BGR_p; @BZ] and in the present paper) is two-folded. Firstly, we look for embeddings of $X$ not into a single Lebesgue space $L_{K}^{q}$ but into a sum of Lebesgue spaces $L_{K}^{q_1}+L_{K}^{q_2}$. This allows to study separately the behaviour of the potentials $V,K$ at $0$ and $\infty$, and to assume independent set of hypotheses about these behaviours. Secondly, we assume hypotheses not on $V$ and $K$ separately but on their ratio, so allowing asymptotic behaviors of general kind for the two potentials.

Thanks to this second novelty we obtain embedding results, and thus existence results for equation , for more general kinds of potentials than the power type ones (cf. Example \[ex2\] below), which are essentially the only ones considered in the existing literature (cf. [@Su-Wang]). Moreover, thanks to the first novelty, we get new results also for power type potentials (cf. Example \[ex2\] below).

This paper is organized as follows. In Section 2 we introduce the hypotheses on $A,V,K$ and the function spaces $D_A$ and $X$ in which we will work. In Section \[COMP\] we state a general result concerning the embedding properties of $X$ into $L_{K}^{q_{1}}+L_{K}^{q_{2}}$ (Theorem \[THM(cpt)\]) and some explicit conditions ensuring that the embedding is compact (Theorems \[THM0\] and \[THM1\]). The general result is proved in Section \[SEC:1\], the explicit conditions in Section \[SEC:2\]. In Section \[SEC: ex\] we apply our embedding results to get existence of non negative solutions for (\[EQ\]). In section \[SEC:EX\] we give some examples to explain the novelty of our results.

We end this introductory section by collecting some notations used in the paper.

$\bullet $ $\mathbb{R}_{+} = ( 0, +\infty ) = \left\{ x\in \mathbb{R} : x>0 \right\}$.

$\bullet $ For every $R>0$, we set $B_{R} =\left\{ x\in \mathbb{R}
^{N}:\left| x\right| <r\right\} $.

$\bullet$ $\omega_N$ is the $(N-1)-$dimensional measure of the unit sphere $\partial B_1 = \left\{ x\in \mathbb{R}
^{N}:\left| x\right| =1 \right\} $.

$\bullet $ For any subset $A\subseteq \mathbb{R}^{N}$, we denote $
A^{c}:=\mathbb{R}^{N}\setminus A$. If $A$ is Lebesgue measurable, $\left|
A\right| $ stands for its measure.

$\bullet $ $C_{\mathrm{c}}^{\infty }(\Omega )$ is the space of the infinitely differentiable real functions with compact support in the open set $\Omega \subseteq \mathbb{R}^{N}$. If $\Omega$ has radial symmetry, $C_{\mathrm{c}, r}^{\infty }( \Omega )$ is the subspace of $C_{\mathrm{c}}^{\infty }(\Omega )$ made of radial functions.

$\bullet $ For any measurable set $A\subseteq \mathbb{R}^{N}$, $
L^{q}(A)$ and $L_{\mathrm{loc}}^{q}(A)$ are the usual real Lebesgue spaces. If $\rho :A\rightarrow \mathbb{R}_{+}$ is a measurable function, then $L^{p}(A,\rho \left( z\right) dz)$ is the real Lebesgue space with respect to the measure $\rho \left( z\right) dz$ ($dz$ stands for the Lebesgue measure on $\mathbb{R}^{N}$). In particular, if $K:\mathbb{R}_{+}\rightarrow \mathbb{R}
_{+} $ is measurable, we denote $L_{K}^{q}\left( A\right) :=L^{q}\left(
A,K\left( \left| x\right| \right) dx\right) $.

$\bullet $ $p^{\prime }:=p/(p-1)$ is the Hölder-conjugate exponent of $p.$

Hypotheses and preliminary results 
===================================

Throughout this paper we assume $N\geq 3$ and $1<p<N$. We will make use of the following hypotheses on $A,V,K$:

-   $A : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} $ is continuous and there exist real numbers $p-N<a_{0},a_{\infty} \leq p$ and $c_{0},c_{\infty}>0$ such that: $$c_{0} \leq \liminf_{r\rightarrow 0^+}\frac{A(r)}{r^{a_{0}}} \leq \limsup_{r\rightarrow 0^+}\frac{A(r)}{r^{a_{0}}} < + \infty,$$ $$c_{\infty}\leq \liminf_{r\rightarrow +\infty}\frac{A(r)}{r^{a_{\infty}}} \leq \limsup_{r\rightarrow +\infty}\frac{A(r)}{r^{a_{\infty}}} <+\infty .$$

-   $V:\mathbb{R}_{+}\rightarrow
    \left[ 0,+\infty \right) $ is measurable and such that $V\in
    L_{\mathrm{loc}}^{1}\left( \mathbb{R}_{+} \right) ;$

-   $K:\mathbb{R}_{+} \rightarrow \mathbb{R}_{+} $ is measurable and such that $K\in L_{\mathrm{loc}}^{s}\left( \mathbb{R}_{+} \right) $ for some $s>1$.

\[remA\] It is easy to check that the hypothesis $\left( \mathbf{A}\right) $ implies that, for each $R>0$, there exist $C_{0}=C_{0}(R)>0$ and $C_{\infty}=C_{\infty}(R)>0$ such that $$\label{remA.1}
A(|x|)\geq C_{0}|x|^{a_{0}} \quad   \mbox{for all} \,\, 0<|x|\leq R,$$ $$\label{remA.2}
A(|x|)\geq C_{\infty}|x|^{a_{\infty}} \quad  \mbox{for all} \,\, |x|\geq R.$$

We now introduce the space functions in which we will work. These are the following:

-   $D_A$ is the closure of $C_{\mathrm{c}, r}^{\infty }( \mathbb{R}^{N} ) $ with respect to the norm $||u||_A := \left( \int_{\mathbb{R}^{N}} A(|x|)|\nabla u|^p 
    dx \right)^{1/p}$ (see also Definition \[DEF:D\_A\] below),

-   $X := D_A \cap  L^{p}(\mathbb{R}^{N}, V (|x|) dx )$ with norm $||u|| := \left( ||u||_{A}^{p} + ||u||_{L^{p}(\mathbb{R}^{N}, V (|x|) dx )}^{p} \right)^{1/p}$.

The rest of this section is devoted to elucidate the characteristics of the functions in $D_A$. In particular we prove some relevant pointwise estimates and embedding results. To be precise, we define

$$S_A := \left\{  u \in C_{\mathrm{c}, r}^{\infty }( \mathbb{R}^{N} )  \, \Big| \, \int_{\mathbb{R}^{N}}A(|x|) |\nabla u |^p dx <+\infty  \right\}.$$

$S_A$ is a linear subspace of $C_{\mathrm{c}, r}^{\infty }( \mathbb{R}^{N} ) $ and $||u||_A = \left( \int_{\mathbb{R}^{N}}A(|x|) |\nabla u |^p dx \right)^{1/p}$ is a norm on it. The next lemmas gives the relevant pointwise estimates for the functions in $S_A$. In all this paper, for any radial function $u$, with a little abuse of notations, we will write $u(x)= u(|x|)= u(r)$ if $r =|x|$.

\[A1\] Assume the hypothesis $\left( \mathbf{A}\right) $. Fix $R_0>0$. Then there exists a constant $C=C(N, R_0, p, a_{\infty} )>0$ such that for all $u \in S_A$ one has $$\label{est1}
|u(x)|\leq C\, |x|^{-\frac{N+a_{\infty}-p}{p}}\, \left( \int_{B^{c}_{R_0}} A(|x|) \, |\nabla u |^p \, dx \right)^{1/p}    \quad \text{for } \;|x|\geq R_0.$$

Assume $u\in S_A$. For $|x|= r\geq R_0$ we have $$-u(r)=\int_{r}^{\infty}u'(s)ds.$$ Using the hypothesis $\left( \mathbf{A}\right) $ and Hölder inequality, we obtain $$\begin{aligned}
|u(r)| &\leq& \int_{r}^{\infty}|u'(s)|ds 
=\int_{r}^{\infty}|u'(s)|s^{\frac{N+a_{\infty}-1}{p}}s^{-\frac{N+a_{\infty}-1}{p}}ds
\\
&\leq&\left(\int_{r}^{\infty}|u'(s)|^{p}s^{N-1}s^{a_{\infty}}ds\right)^{\frac{1}{p}}\left(\int_{r}^{\infty}s^{-\frac{{N+a_{\infty}-1}}{p-1}} ds\right)^{\frac{p-1}{p}}
\\
&=&  (\omega_{N})^{-\frac{1}{p}}\left(\int_{B_{r}^c}|x|^{a_{\infty}}|\nabla u|^{p}dx\right)^{\frac{1}{p}}\left(\int_{r}^{\infty} s^{-\frac{{N+a_{\infty}-1}}{p-1}} ds\right)^{\frac{p-1}{p}}
\\
&\leq&  C_{\infty} ^{-1/p} \omega_{N}^{-\frac{1}{p}}\left( \frac{p-1}{a_{\infty} +N-p}    \right)^{\frac{p-1}{p}}  \left(\int_{B_{r}^c} A(|x|)|\nabla u|^{p}dx\right)^{\frac{1}{p}}\, r^{-\frac{{N+a_{\infty}-p}}{p}},\end{aligned}$$ where $C_{\infty}=C_{\infty}(R_0)$ is the constant in \[remA.2\]. Hence the thesis follows.

\[A2\] Assume the hypothesis $\left( \mathbf{A}\right) $. Fix $R_0 >0$. Then there exists a constant $C= C(N, R_0, p, a_0 )>0$ such that for all $u\in S_A \cap C_{c,r}^{\infty }(B_{R_0})$ one has $$|u(x)|\leq C\, |x|^{-\frac{N+a_{0}-p}{p}}\, \left( \int_{B_{R_0}} A(|x|) \, |\nabla u |^p \, dx \right)^{1/p} 
\quad \text{for }0<|x|<R_0.$$

Let $u\in S_A \cap C_{c,r}^{\infty }(B_{R_0})$ and take $|x|=r<R_0$. Since $u(R_0 )=0$, we have $$-u(r)=u(R_0)-u(r)=\int_{r}^{R_0}u'(s)ds.$$ The same arguments of Lemma \[A1\] yield $$\begin{aligned}
|u(r)|&\leq&\int_{r}^{R_0}|u'(s)|ds
\leq\left(\int_{r}^{R_0}|u'(s)|^{p}s^{N-1}s^{a_{0}}ds\right)^{\frac{1}{p}}\left(\int_{r}^{R_0}  s^{-\frac{N+a_{0}-1}{p-1}}  ds\right)^{\frac{p-1}{p}}
\\
&\leq &(\omega_{N})^{-\frac{1}{p}} \left(\int_{B_{R_0} } |x|^{a_0}|\nabla u|^{p}dx\right)^{\frac{1}{p}} \left(\int_{r}^{R_0}s^{-\frac{N+a_{0}-1}{p-1}} ds\right)^{\frac{p-1}{p}}
\\
&\leq & \omega_{N}^{-\frac{1}{p}}  C_0^{-1/p}\, \left(\frac{p-1}{N+a_{0}-p} \right)^{p-1/p} \left( \int_{B_{R_0}} A(|x|) \, |\nabla u |^p \, dx \right)^{1/p} \,   r^{-\frac{N+a_{0}-p}{p}},\end{aligned}$$ where $C_0=C_0 (R_0)$ is the constant in \[remA.1\]. Then the thesis ensues.

\[ALem\] Assume the hypothesis $\left( \mathbf{A}\right) $. Fix $R_0 >0$. Then there exists a constant $C= C(N, R_0, p,a_0 , a_{\infty})>0$ such that for all $u\in   S_A$ one has $$|u(x)|\leq C\, |x|^{-\frac{N+a_{0}-p}{p}}\, \left( \int_{B_{R_0 +1 }} A(|x|) \, |\nabla u |^p \, dx + \int_{B^{c}_{R_0  }} A(|x|) | \nabla u|^p  dx      \right)^{1/p} \quad \text{for } 0<|x|<R_0.$$

Let $u\in   S_A$. Take a radial function $\rho  \in    C_{c,r}^{\infty}(\mathbb{R}^{N})$ such that $ \rho (x) \in [0,1]$, $\rho \equiv 1 $ in $B_{R_0} $ and $\rho (x) \equiv 0$ if $|x| \geq R_0 + 1/2$. Hence $\rho \, u \in C_{c,r}^{\infty}(B_{R_0 +1})$, so that we can apply Lemma \[A2\] (in the ball $B_{R_0 +1} )$ and get $$| \rho (x) u (x) | \leq C |x|^{-\frac{N+ a_0 -p}{p}} \left( \int_{B_{R_0 +1}} A(|x|) |\nabla (\rho  u)|^p dx \right)^{1/p}.$$ If $|x| < R_0 $ we have $\rho (x) =1$ and hence $$| u (x) | \leq C |x|^{-\frac{N+ a_0 -p}{p}} \left( \int_{B_{R_0 +1}} A(|x|) |\nabla  (\rho u)|^p dx \right)^{1/p}.$$ We also have $$|\nabla  (\rho u)|^p \leq \left( \rho |\nabla u | + |u| |\nabla \rho | \right)^p \leq C_p \left(\rho^p |\nabla u|^p + |u|^p |\nabla \rho |^p \right)$$ and hence, for $x \in B_{R_0}$, $$| u (x) | \leq C |x|^{-\frac{N+ a_0 -p}{p}} C_p^{1/p}\left( \int_{B_{R_0 +1}} A(|x|) |\nabla  u|^p dx + \int_{B_{R_0 +1}} A(|x|) |u|^p \,|\nabla  \rho|^p dx\right)^{1/p}.$$ Moreover $$\int_{B_{R_0 +1}} A(|x|) |u|^p \,|\nabla  \rho|^p dx \leq C_1 \int_{B_{R_0 +1}\backslash B_{R_0}} A(|x|) |u|^p \, dx$$ where the constant $C_1= \max |\nabla \rho |^p $ depends only on $\rho$, hence on $R_0$. We can now apply Lemma \[A1\] in the domain $B^{c}_{R_0}$, and we get $$|u|^p (y) \leq C |y|^{-N- a_{\infty} +p} \int_{B^{c}_{R_0 }} A(|x|) |\nabla  u|^p dx \quad \text{for } |y|> R_0.$$ Recalling that $A$ is bounded in $B_{R_0 +1}\backslash B_{R_0} $, for $y \in B_{R_0 +1}\backslash B_{R_0} $ and $C_2 = 
\max \left\{ A(|y|) \, \big| \, y\in B_{R_0 +1}\backslash B_{R_0} \right\}$ we get $$A(|y|) |u|^p (y)  \leq C_2 \, C |y|^{-N - a_{\infty} +p} \int_{B^{c}_{R_0 }} A(|x|) |\nabla  u|^p dx$$ and hence, integrating w.r.t. $y \in B_{R_0 +1}\backslash B_{R_0} $, we obtain $$\int_{B_{R_0 +1}\backslash B_{R_0}}A(|y|) |u|^p (y) dy \leq C_3 \int_{B^{c}_{R_0 }} A(|x|) |\nabla  u|^p dx$$ where $C_3=C_3 (N, a_{\infty}, R_0 , p)$. Pasting all together, for $|x|<R_0$ we get $$\begin{aligned}
| u (x) | &\leq &C |x|^{-\frac{N+ a_0 -p}{p}} C_p^{1/p}\left( \int_{B_{R_0 +1}} A(|x|) |\nabla  u|^p dx + \int_{B_{R_0 +1}} A(|x|) |u|^p \,|\nabla  \rho|^p dx\right)^{1/p} 
\\
&\leq &C_4 |x|^{-\frac{N+ a_0 -p}{p}} \left( \int_{B_{R_0 +1}} A(|x|) |\nabla  u|^p dx + \int_{B^{c}_{R_0 }} A(|x|) \,|\nabla  u|^p dx \right)^{1/p}, \end{aligned}$$ where $C_4 =C_4 (N, R_0 , p, a_0, a_{\infty})$. Hence the thesis.

We can now give a precise definition of $D_A$.

\[DEF:D\_A\] $D_A$ is the completion of $S_A$ with respect to the norm $|| \cdot ||_A$.

The pointwise estimates of the previous lemmas imply the following proposition, which gives the main properties of $D_A$. The proof is a simple exercise in funtional analysis and we skip it.

\[lemdens2\]

Assume the hypothesis $\left( \mathbf{A}\right) $. Then the following properties hold.

-   If $u \in D_A$, then $u$ has weak derivatives $D_i u$ in the open set $\Omega=\mathbb{R}^{N} \backslash \{0 \}$ ($i=1,..., N$) and one has $D_i u \in L^p_{loc}(\Omega )$.

-   If $u \in D_A$, then $\int_{\mathbb{R}^{N}} A(|x|) |\nabla u|^p \, dx < + \infty $ and $||u||_A =\left( \int_{\mathbb{R}^{N}} A(|x|) |\nabla u|^p \, dx
        \right)^{1/p}$ is a norm on $D_A$. With this norm, $D_A$ is a Banach space.

-   For any $R_0 >0$, there exists a constant $M=M(N, R_0 , a_0 , a_{\infty})>0$ such that for all $u \in D_A$ one has

    $$|u(x) | \leq M |x|^{-\frac{N+ a_0 -p}{p}} \, ||u||_A , \quad \mbox{for a.e.}\, x \in B_{R_0},$$

    $$|u(x) | \leq M |x|^{-\frac{N+ a_{\infty} -p}{p}} \, ||u||_A ,\quad \mbox{for a.e.}\,  x \in B^{c}_{R_0}.$$

We now prove some embedding properties of the space $D_A$. To this aim, we define the exponents $p_{0}, p_{\infty}$ as follows: $$p_{0}:=\frac{pN}{N+a_{0}-p}\,,\quad p_{\infty}:=\frac{pN}{N+a_{\infty}-p}.$$ Recall that $p-N<a_{0},\, a_{\infty}\leq p$ and notice that, from the hypotheses, we have $p_{0}, p_{\infty}\geq p$.

\[A4\]

Assume the hypothesis $\left( \mathbf{A}\right) $. For every $R >0$ we have the continuous embeddings $$D_A \hookrightarrow L^{p_{0}}(B_{R}) \quad \text{and} \quad D_A \hookrightarrow L^{p_{\infty}}(B^{c}_{R}).$$

Let $u\in    S_A$ (the general case follows by density). Fix $R>0$ and denote by $C$ any positive constant only dependent on $N$, $p$, $a_0$, $a_\infty$ and $R$. We first prove the embedding in $L^{p_{\infty}}(B^{c}_{R})$, so we estimate the norm of $u$ in such a space. With an integration by parts, we get

$$\int_{B_{R}^c} |u(x)|^{p_{\infty}} dx = \omega_N \int_{R}^{+\infty} r^{N-1} |u (r)|^{p_{\infty}}dr \leq \frac{\omega_N p_{\infty} }{N}
\int_{R}^{+\infty} r^N |u (r)|^{p_{\infty}-1}\, |u' (r) | dr.$$

Then, by several applications of Hölder inequality and using the pointwise estimates of Lemma \[lemdens2\], we get

$$\int_{R}^{+\infty} r^N |u (r)|^{p_{\infty}-1} \, |u' (r) | dr = \int_{R}^{+\infty} r^{\frac{N-1}{p}} |u' (r)| r^{\frac{a_{\infty}}{p}} \,
r^{-\frac{a_{\infty}}{p}} r^{\frac{Np-N+1}{p}} |u (r)|^{p_{\infty}-1} dr$$ $$\leq \left( \int_{R}^{+\infty} r^{N-1}  |u' (r)|^p r^{a_{\infty}} dr \right)^{1/p} \, \left( \int_{R}^{+\infty} r^{\frac{Np-N+1-a_{\infty}}{p-1}}
|u (r)|^{(p_{\infty}-1)\frac{p}{p-1} }dr \right)^{\frac{p-1}{p}}$$

$$\leq C\left(\int_{R}^{+\infty}  r^{N-1} A(r) |u' (r)|^p   dr  \right)^{1/p} \, \left( \int_{R}^{+\infty} r^{N-1} |u(r)|^{p_{\infty}} \,
r^{\frac{p-a_{\infty}}{p-1}} |u(r)|^{\frac{p_{\infty} -p}{p-1}} dr   \right)^{\frac{p-1}{p}}$$

$$\leq C \left( \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx  \right)^{1/p} \times$$ $$\times \left(\int_{R}^{+\infty} r^{N-1} |u(r)|^{p_{\infty}} 
r^{\frac{p-a_{\infty}}{p-1} } \, r^{-\frac{N+a_{\infty}-p}{p} \frac{p_{\infty}-p}{p-1} } \, \left(  \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx \right)^{\frac{p_{\infty}-p}{p(p-1)}}  \right)^{\frac{p-1}{p}}$$

$$\leq C \left( \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx  \right)^{\frac{p_{\infty}}{p^2}} \, \left( \int_{B_{R}^c} |u(x)|^{p_{\infty}} dx\right)^{\frac{p-1}{p}}$$

Notice that we have $\frac{p-a_{\infty}}{p-1} - \frac{N+a_{\infty}-p}{p} \frac{p_{\infty}-p}{p-1} =0  $, from the definition of $p_{\infty}$.

From the previous computations, we get

$$\int_{B_{R}^c} |u(x)|^{p_{\infty}} dx \leq C \left( \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx  \right)^{\frac{p_{\infty}}{p^2}} \, \left( \int_{B_{R}^c} |u(x)|^{p_{\infty}} dx\right)^{\frac{p-1}{p}}$$

and hence

$$\left( \int_{B_{R}^c} |u(x)|^{p_{\infty}} dx  \right)^{1/p} \leq C \left( \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx  \right)^{\frac{p_{\infty}}{p^2}} ,$$

that is

$$\left( \int_{B_{R}^c} |u(x)|^{p_{\infty}} dx  \right)^{\frac{1}{p_{\infty}}} \leq C \left( \int_{B_{R}^c} A(|x|) \, |\nabla u (x)|^p dx  \right)^{\frac{1}{p}} \leq
C ||u||_A ,$$

which is the thesis. In order to prove the embedding in $L^{p_{0}}(B_{R})$, we use an argument similar to the one of Lemma \[ALem\]. So we fix a cut-off function $\rho$ as we did there (with $R$ instead of $R_0$). For $u \in S_A$ we set $v= \rho u \in C_{c,r}^{\infty}(B_{R+1})$. Arguing as for the previous case, we get

$$\int_{B_{R+1}} |v(x)|^{p_0} dx = \omega_N \int_{0}^{R+1} r^{N-1} |v(r)|^{p_0}  dr \leq \frac{p_0 \omega_N}{N} \int_{0}^{R+1} r^{N} |v(r)|^{p_0 -1} |v' (r)| dr=$$

$$= \frac{p_0 \omega_N}{N} \int_{0}^{R+1}  r^{\frac{N-1}{p}} |v'(r)| r^{\frac{a_0}{p}}r^{-\frac{a_0}{p}} r^{\frac{Np-N+1}{p}} |v(r)|^{p_0 -1} dr$$

$$\leq \frac{p_0 \omega_N}{N} \left( \int_{0}^{R+1} r^{N-1} |v' (r)|^p r^{a_0} dr \right)^{1/p} \left( \int_{0}^{R+1} r^{\frac{Np-N+1-a_0}{p-1}} 
|v(r)|^{(p_0 -1)\frac{p}{p-1}} dr \right)^{\frac{p-1}{p}}$$

$$\leq  \frac{p_0 \omega_N}{N} \left( \int_{0}^{R+1} r^{N-1} A(r) |v' (r)|^p dr \right)^{1/p} \left( \int_{0}^{R+1} r^{N-1} |v(r)|^{p_0}
r^{\frac{p-a_0}{p-1}} |v(r)|^{\frac{p_0 -p}{p-1}} dr \right)^{\frac{p-1}{p}}$$

$$\leq C \left( \int_{B_{R+1}} A(|x|) |\nabla v (x)|^p dx \right)^{1/p} \times$$ $$\times \left( \int_{0}^{R+1} r^{N-1} |v(r)|^{p_0} r^{\frac{p-a_0}{p-1}}  r^{-\frac{N+a_{0}-p}{p} \frac{p_{0}-p}{p-1} } \, \left(  \int_{B_{R+1}} A(|x|) \, |\nabla v (x)|^p dx \right)^{\frac{p_{0}-p}{p(p-1)}} \right)^{\frac{p-1}{p}}$$

$$=  C \left( \int_{B_{R+1}} A(|x|) |\nabla v (x)|^p dx \right)^{\frac{p_0}{p^2}}  \left( \int_{B_{R+1}} |v(x)|^{p_0} dx \right)^{\frac{p-1}{p}} ,$$

Notice that $\frac{p-a_0}{p-1} -\frac{N+a_{0}-p}{p} \frac{p_{0}-p}{p-1} =0$, by the definition of $p_0$. From these computations we easily deduce, as before, that

$$\left( \int_{B_{R+1}}  |v (x)|^{p_0} dx \right)^{\frac{1}{p_0}}  \leq C \left( \int_{B_{R+1}} A(|x|) |\nabla v (x)|^p dx \right)^{\frac{1}{p}}.$$

Now, as in Lemma \[ALem\], we use the fact that $A$ is continuous and strictly positive on the compact set ${\overline {B_{R+1} \backslash B_{R} }}$, and we get

$$\int_{B_{R+1}} A(|x|) |\nabla  \rho u|^p dx \leq C \int_{B_{R+1}} A(|x|) \rho^p |\nabla   u|^p dx + C \int_{B_{R+1} \backslash B_{R}} A(|x|) |u|^p |\nabla \rho |^p dx
\leq$$

$$\leq C ||u||^{p}_A + C \int_{B_{R}^c} A(|x|) | \nabla u|^p dx \left( \int_{B_{R+1} \backslash B_{R}} x^{-N-a_{\infty}+p}dx \right)^{1/p} \leq C ||u||^{p}_A .$$

Hence we get

$$\left( \int_{B_{R+1}} |\rho u|^{p_0} dx \right)^{1/p_0} \leq C ||u||_A$$

and therefore

$$\left( \int_{B_{R}} | u|^{p_0} dx \right)^{1/p_0} = \left( \int_{B_{R}} |\rho u|^{p_0} dx \right)^{1/p_0} \leq \left( \int_{B_{R+1}} |\rho u|^{p_0} dx 
\right)^{1/p_0} \leq C ||u||_A ,$$

which is the thesis.

The following lemma gives another embedding result that we will use.

\[A6\]

Assume the hypothesis $\left( \mathbf{A}\right) $ and fix $0<r<R$. Then the embedding $$D_{A} \hookrightarrow L^{p}(B_R \setminus {\overline B_r })$$ is continuous and compact.

Set $E:=B_R \setminus {\overline B_r }$ for brevity. The continuity of the embedding easily derives from (\[est1\]), by integrating over the set $E$. As to compactness, let $\{u_{n}\}_n$ be a bounded sequence in $D_{A} $. By continuity of the embedding we obtain that also $\{|| u_n  ||_{L^{p}(E)} \}_n$ is bounded. Moreover, as $A$ is continuous and strictly positive on the compact set $\overline {E }$, we have

$$\int_{E} |\nabla u_n |^p dx \leq C \int_{ E } A(|x|) |\nabla u_n |^p dx
 \leq C_1 .$$

Thus $\{u_{n}\}_n$ is bounded also in the space $W^{1,p}(E )$. Thanks to Rellich’s Theorem, $\{u_{n}\}_n$ has a convergent subsequence in $L^{p}(E )$, and this gives the thesis.

Compactness results for the space $X$ {#COMP}
=====================================

In this section we state the main compactness results of this paper, concerning the space $X$. Recall that we define such a space as $$X := D_A \cap  L^{p}(\mathbb{R}^{N}, V (|x|) dx )$$ endowed with the norm $||u|| := \left( ||u||_{A}^{p} + ||u||_{L^{p}(\mathbb{R}^{N}, V (|x|) dx )}^{p} \right)^{1/p}$, with respect to which $X$ is a Banach space. The compactness results that we state here will be proved in sections \[SEC:1\] and \[SEC:2\].

Given $A$, $V$ and $K$ as in $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $ and $\left( \mathbf{K}\right) $, we define the following functions of $R>0$ and $q>1$: $$\begin{aligned}
\mathcal{S}_{0}\left( q,R\right)&:=&
\sup_
{u\in X,\,\left\| u\right\| =1  }
\int_{B_{R}}K\left( \left| x\right| \right)
\left| u\right| ^{q}dx,  \label{S_o :=}
\\
\mathcal{S}_{\infty }\left( q,R\right)&:=&
\sup_
{u\in X,\,\left\| u\right\| =1  } 
\int_{\mathbb{R}^{N}\setminus B_{R}}K\left( \left| x\right| \right) \left| u\right| ^{q}dx.
\label{S_i :=}\end{aligned}$$ Clearly $\mathcal{S}_{0}\left( q,\cdot \right) $ is nondecreasing, $\mathcal{
S}_{\infty }\left( q,\cdot \right) $ is nonincreasing and both of them can be infinite at some $R$.

Our first result concerns the embedding properties of $X$ into the sum space $$L_{K}^{q_{1}}+L_{K}^{q_{2}}:=\left\{ u_{1}+u_{2}:u_{1}\in
L_{K}^{q_{1}}\left( \mathbb{R}^{N}\right) ,\,u_{2}\in L_{K}^{q_{2}}\left( \mathbb{R}^{N}\right) \right\} ,\quad 1<q_{i}<\infty .$$ We recall from [@BPR] that such a space can be characterized as the set of measurable mappings $u:\mathbb{R}^{N}\rightarrow \mathbb{R}$ for which there exists a measurable set $E\subseteq \mathbb{R}^{N}$ such that $u\in
L_{K}^{q_{1}}\left( E\right) \cap L_{K}^{q_{2}}\left( E^{c}\right) $. It is a Banach space with respect to the norm $$\left\| u\right\| _{L_{K}^{q_{1}}+L_{K}^{q_{2}}}:=\inf_{u_{1}+u_{2}=u}\max
\left\{ \left\| u_{1}\right\| _{L_{K}^{q_{1}}(\mathbb{R}^{N})},\left\|
u_{2}\right\| _{L_{K}^{q_{2}}(\mathbb{R}^{N})}\right\}$$ and the continuous embedding $L_{K}^{q}\hookrightarrow
L_{K}^{q_{1}}+L_{K}^{q_{2}}$ holds for all $q\in \left[ \min \left\{
q_{1},q_{2}\right\} ,\max \left\{ q_{1},q_{2}\right\} \right] $. The assumptions of our result are quite general but not so easy to check, so more handy conditions ensuring these general assumptions will be provided by the next results.

\[THM(cpt)\] Let $1<p<N$, let $A$, $V$ and $K$ be as in $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $ and $\left( \mathbf{K}\right) $, and let $q_{1},q_{2}>1$.

-   If $$\mathcal{S}_{0}\left( q_{1},R_{1}\right) <\infty \quad \text{and}\quad 
    \mathcal{S}_{\infty }\left( q_{2},R_{2}\right) <\infty \quad \text{for some }R_{1},R_{2}>0,  
    \tag*{$\left( {\cal S}_{q_{1},q_{2}}^{\prime }\right) $}$$ then $X$ is continuously embedded into $L_{K}^{q_{1}}(\mathbb{R}^{N})+L_{K}^{q_{2}}(\mathbb{R}^{N})$.

-   If $$\lim_{R\rightarrow 0^{+}}\mathcal{S}_{0}\left( q_{1},R\right)
    =\lim_{R\rightarrow +\infty }\mathcal{S}_{\infty }\left( q_{2},R\right) =0, 
    \tag*{$\left({\cal S}_{q_{1},q_{2}}^{\prime \prime }\right) $}$$ then $X$ is compactly embedded into $L_{K}^{q_{1}}(\mathbb{R}^{N})+L_{K}^{q_{2}}(\mathbb{R}^{N})$.

Observe that, of course, $(\mathcal{S}_{q_{1},q_{2}}^{\prime \prime })$ implies $(\mathcal{S}_{q_{1},q_{2}}^{\prime })$. Moreover, these assumptions can hold with $q_{1}=q_{2}=q$ and therefore Theorem \[THM(cpt)\] also concerns the embedding properties of $X$ into $L_{K}^{q}$, $1<q<\infty $.

We now look for explicit conditions on $V$ and $K$ implying $(\mathcal{S}_{q_{1},q_{2}}^{\prime \prime })$ for some $q_{1}$ and $q_{2}$. More precisely, in Theorem \[THM0\] we will find a range of exponents $q_{1} $ such that $\lim_{R\rightarrow 0^{+}}\mathcal{S}_{0}\left(q_{1},R\right)$ $=0$, while in Theorem \[THM1\] we will do the same for exponents $q_{2}$ such that $\lim_{R\rightarrow +\infty}\mathcal{S}_{\infty }\left( q_{2},R\right) =0$.

For $\alpha \in \mathbb{R}$, $\beta \in \left[ 0,1\right] $, $a>p-N$, we define two functions $\alpha ^{*}\left(a, \beta \right) $ and $q^{*}\left(a, \alpha ,\beta
\right) $ by setting $$\alpha ^{*}\left(a, \beta \right) :=\max \left\{ p\beta -1-\frac{p-1}{p}N -a \beta + \frac{a}{p},-\left(
1-\beta \right) N\right\} =$$ $$\left\{ 
\begin{array}{ll}
p\beta -1-\frac{p-1}{p}N  -a\beta +\frac{a}{p}\quad \smallskip & \text{if }0\leq \beta \leq \frac{1}{p} \\ 
-\left( 1-\beta \right) N & \text{if }\frac{1}{p}\leq \beta \leq 1
\end{array}
\right.$$ and $$q^{*}\left(a, \alpha ,\beta \right) :=p\frac{\alpha -p\beta +N+ a \beta}{N-p+a}.$$

\[THM0\] Let $A$, $V$, $K$ be as in $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $, $\left( \mathbf{K}\right) $. Assume that there exists $R_{1}>0$ such that $V\left( r\right) <+\infty $ almost everywhere in $(0,R_1)$ and $$\operatorname*{ess\,sup}_{r\in \left( 0,R_{1}\right) }\frac{K\left( r\right) }{r^{\alpha _{0}}V\left( r\right) ^{\beta _{0}}}<+\infty \quad \text{for some }0\leq \beta _{0}\leq 1\text{~and }\alpha _{0}>\alpha ^{*}\left(a_{0}, \beta
_{0}\right) .  \label{esssup in 0}$$ Then $\displaystyle \lim_{R\rightarrow 0^{+}}\mathcal{S}_{0}\left(
q_{1},R\right) =0$ for every $q_{1}\in \mathbb{R}$ such that $$\max \left\{ 1,p\beta _{0}\right\} <q_{1}<q^{*}\left(a_{0}, \alpha _{0},\beta
_{0}\right) .  \label{th1}$$

\[THM1\] Let $A$, $V$, $K$ be as in $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $, $\left( \mathbf{K}\right) $. Assume that there exists $R_{2}>0$ such that $V\left( r\right) <+\infty $ for almost every $r>R_2$ and $$\operatorname*{ess\,sup}_{r>R_{2}}\frac{K\left( r\right) }{r^{\alpha _{\infty
}}V\left( r\right) ^{\beta _{\infty }}}<+\infty \quad \text{for some }0\leq
\beta _{\infty }\leq 1\text{~and }\alpha _{\infty }\in \mathbb{R}.
\label{esssup all'inf}$$ Then $\displaystyle \lim_{R\rightarrow +\infty }\mathcal{S}_{\infty }\left(
q_{2},R\right) =0$ for every $q_{2}\in \mathbb{R}$ such that $$q_{2}>\max \left\{ 1,p\beta _{\infty },q^{*}\left(a_{\infty}, \alpha _{\infty },\beta
_{\infty }\right) \right\} .  \label{th2}$$

We observe explicitly that for every $a$, $\alpha$, $\beta $ as above one has $$\max \left\{ 1,p\beta ,q^{*}\left(a, \alpha ,\beta \right) \right\} =\left\{ 
\begin{array}{ll}
q^{*}\left(a, \alpha ,\beta \right) \quad & \text{if }\alpha \geq \alpha
^{*}\left(a, \beta \right) \smallskip \\ 
\max \left\{ 1,p\beta \right\} & \text{if }\alpha \leq \alpha ^{*}\left(a,
\beta \right)
\end{array}
\right. .$$

\[RMK: suff12\]

1.  \[RMK: suff12-V\^0\]We mean $V\left( r\right) ^{0}=1$ for every $r$ (even if $V\left( r\right) =0$). In particular, if $V\left( r\right) =0$ for almost every $r>R_{2}$, then Theorem \[THM1\] can be applied with $\beta
    _{\infty }=0$ and assumption (\[esssup all’inf\]) means $$\operatorname*{ess\,sup}_{r>R_{2}}\frac{K\left( r\right) }{r^{\alpha _{\infty }}}<+\infty \quad \text{for some }\alpha _{\infty }\in \mathbb{R}.$$ Similarly for Theorem \[THM0\] and assumption (\[esssup in 0\]), if $V\left( r\right) =0$ for almost every $r\in \left( 0,R_{1}\right) $.

2.  \[RMK: suff12-no hp\]The inequality $\max \left\{ 1,p\beta
    _{0}\right\} <q^{*}\left(a_0 , \alpha _{0},\beta _{0}\right) $ is equivalent to $\alpha _{0}>\alpha ^{*}\left(a_0 , \beta _{0}\right) $. Then, in (\[th1\]), such inequality is automatically true and does not ask for further conditions on $\alpha _{0}$ and $\beta _{0}$.

3.  \[RMK: suff12-Vbdd\]The assumptions of Theorems \[THM0\] and \[THM1\] may hold for different pairs $\left( \alpha _{0},\beta_{0}\right)$, $\left( \alpha _{\infty },\beta _{\infty }\right) $ (assuming $p$ and $a_0$ fixed). In this case, of course, one chooses them in order to get the ranges for $q_{1},q_{2}
    $ as large as possible. For instance, assume that $a_0 \leq p$ and $V$ is not singular at the origin, i.e., $V$ is essentially bounded in a neighbourhood of 0. If condition (\[esssup in 0\]) holds true for a pair $\left( \alpha _{0},\beta _{0}\right) $, then (\[esssup in 0\]) also holds for all pairs $\left( \alpha
    _{0}^{\prime },\beta _{0}^{\prime }\right) $ such that $\alpha _{0}^{\prime
    }>\alpha _{0}$ and $\beta _{0}^{\prime }<\beta _{0}$. Therefore, since $\max
    \left\{ 1,p\beta \right\} $ is nondecreasing in $\beta $ and $q^{*}\left(a, 
    \alpha ,\beta \right) $ is increasing in $\alpha $ and decreasing in $\beta $ (because $a_0 \leq p$) , it is convenient to choose $\beta _{0}=0$ and the best interval where one can take $q_{1}$ is $1<q_{1}<q^{*}\left(a_0 , \overline{\alpha },0\right) $ with $\overline{\alpha }:=\sup \left\{ \alpha _{0}:\operatorname*{ess\,sup}_{r\in \left(
    0,R_{1}\right) }K\left( r\right) /r^{\alpha _{0}}<+\infty \right\} $ (here we mean $q^{*}\left(a_0 , +\infty ,0\right) =+\infty $).

Proof of Theorem \[THM(cpt)\] \[SEC:1\]
=======================================

Assume as usual $N\geq 3$ and $1<p<N$, and let $A$, $V$ and $K$ be as in $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $ and $\left( \mathbf{K}\right) $. Recall from assumption $\left( \mathbf{K}\right) $ that $K\in L_{\mathrm{loc}}^{s}\left( \left( 0,+\infty \right) \right) $ for some $s>1$.

\[Lem(corone)\]Let $R>r>0$ and $1<q<\infty $. Then there exist $\tilde{C}=\tilde{C}\left(N,p,r,R,q,s\right) >0$ and $l=l\left(p,q,s\right) >0$ such that $q-lp>0$ and $\forall u\in X$ one has $$\int_{B_{R}\setminus B_{r}}K\left( \left| x\right| \right) \left|u\right| ^q dx
\leq
\tilde{C}\left\| K\left( \left| \cdot\right| \right) \right\| _{L^{s}(B_{R}\setminus B_{r})}
\left\|u\right\|^{q-lp} \left(\int_{B_R\setminus B_r}\left|u\right|^{p}dx\right)^l.
\label{LEM:corone}$$

Notice that, in the second part of Lemma \[LEM:corone\], $s>\frac{Np}{N(p-1)+p-a_+}$ implies $\tilde{q}>1$.

Let $u\in X$ and fix $t\in(1,s)$ such that $t'q>p$ (where $t'=t/(t-1)$). Then, by Hölder inequality and the pointwise estimates of Section 2, we have $$\begin{aligned}
&& \int_{B_{R}\setminus B_{r}}K\left( \left| x\right| \right) \left| u\right|^{q} dx \\
&\leq & \left( \int_{B_{R}\setminus B_{r}}K\left(\left| x\right| \right) ^{t}dx\right) ^{\frac{1}{t}}
\left( \int_{B_{R}\setminus B_{r}}\left|u\right| ^{t^{\prime }q}dx\right) ^{\frac{1}{t^{\prime }}} \\
&\leq & 
\left| B_{R}\setminus B_{r}\right| ^{\frac{1}{t}-\frac{1}{s}}
\left\| K\left( \left| \cdot \right| \right) \right\|_{L^{s}(B_{R}\setminus B_{r})}
\left( \int_{B_{R}\setminus B_{r}}\left|u\right| ^{t^{\prime }q-p}\left|u\right|^{p} dx\right) ^{\frac{1}{t^{\prime }}}
\\
&\leq &
\left| B_{R}\setminus B_{r}\right| ^{\frac{1}{t}-\frac{1}{s}}
\left\| K\left( \left| \cdot \right| \right) \right\|_{L^{s}(B_{R}\setminus B_{r})}
\left( \frac{C \left\|u\right\| }{r^{\frac{N-p+a_0}{p}}}\right) ^{q-p/t^{\prime}}
\left( \int_{B_{R}\setminus B_{r}}\left|u\right|^{p} dx\right) ^{\frac{1}{t^{\prime }}}.\end{aligned}$$ This proves (\[LEM:corone\]). We now prove Theorem \[THM(cpt)\]. Recall the definitions (\[S\_o :=\])-(\[S\_i :=\]) of the functions $\mathcal{S}_{0}$ and $\mathcal{S}_{\infty }$, and the following result from [@BPR] concerning convergence in the sum of Lebesgue spaces.

\[Prop(-&gt;0)\] Let $\left\{ u_{n}\right\}
\subseteq L_{K}^{p_{1}}+L_{K}^{p_{2}}$ be a sequence such that $\forall
\varepsilon >0$ there exist $n_{\varepsilon }>0$ and a sequence of measurable sets $E_{\varepsilon ,n}\subseteq \mathbb{R}^{N}$ satisfying $$\forall n>n_{\varepsilon },\quad \int_{E_{\varepsilon ,n}}K\left( \left|
x\right| \right) \left| u_{n}\right| ^{p_{1}}dx+\int_{E_{\varepsilon
,n}^{c}}K\left( \left| x\right| \right) \left| u_{n}\right|
^{p_{2}}dx<\varepsilon .  \label{Prop(->0): cond}$$ Then $u_{n}\rightarrow 0$ in $L_{K}^{p_{1}}+L_{K}^{p_{2}}$.

We prove each part of the theorem separately.

\(i) By the monotonicity of $\mathcal{S}_{0}$ and $\mathcal{S}_{\infty }$, it is not restrictive to assume $R_{1}<R_{2}$ in hypothesis $\left( \mathcal{S}_{q_{1},q_{2}}^{\prime }\right) $. In order to prove the continuous embedding, let $u\in X$, $u\neq 0$. Then we have $$\int_{B_{R_{1}}}K\left( \left| x\right| \right) \left| u\right|
^{q_{1}}dx=\left\| u\right\| ^{q_{1}}\int_{B_{R_{1}}}K\left( \left| x\right|
\right) \frac{\left| u\right| ^{q_{1}}}{\left\| u\right\| ^{q_{1}}}dx\leq
\left\| u\right\| ^{q_{1}}\mathcal{S}_{0}\left( q_{1},R_{1}\right) 
\label{pf1}$$ and, similarly, $$\int_{B_{R_{2}}^{c}}K\left( \left| x\right| \right) \left| u\right|
^{q_{2}}dx\leq \left\| u\right\| ^{q_{2}}\mathcal{S}_{\infty }\left(
q_{2},R_{2}\right) .  \label{pf2}$$ We now use (\[LEM:corone\]) of Lemma \[Lem(corone)\] and Lemma \[A6\] to deduce that there exists a constant $C_{1}>0$, independent from $u$, such that $$\int_{B_{R_{2}}\setminus B_{R_{1}}}K\left( \left| x\right| \right) \left|
u\right| ^{q_{1}}dx\leq C_{1}\left\| u\right\| ^{q_{1}}.  \label{pf3}$$ Hence $u\in L_{K}^{q_{1}}(B_{R_{2}})\cap L_{K}^{q_{2}}(B_{R_{2}}^{c})$ and thus $u\in L_{K}^{q_{1}}+L_{K}^{q_{2}}$. Moreover, if $u_{n}\rightarrow 0$ in $X$, then, using (\[pf1\]), (\[pf2\]) and (\[pf3\]), we get $$\int_{B_{R_{2}}}K\left( \left| x\right| \right) \left| u_{n}\right|
^{q_{1}}dx+\int_{B_{R_{2}}^{c}}K\left( \left| x\right| \right) \left|
u_{n}\right| ^{q_{2}}dx=o\left( 1\right) _{n\rightarrow \infty },$$ which means $u_{n}\rightarrow 0$ in $L_{K}^{q_{1}}+L_{K}^{q_{2}}$ by Proposition \[Prop(-&gt;0)\]. **

\(ii) Assume hypothesis $\left( \mathcal{S}_{q_{1},q_{2}}^{\prime
\prime }\right) $. Let $\varepsilon >0$ and let $u_{n}\rightharpoonup 0$ in $
X$. Then $\left\{ \left\| u_{n}\right\| \right\}_n $ is bounded and, arguing as for (\[pf1\]) and (\[pf2\]), we can take $
r_{\varepsilon }>0$ and $R_{\varepsilon }>r_{\varepsilon }$ such that for all $n$ one has $$\int_{B_{r_{\varepsilon }}}K\left( \left| x\right| \right) \left|
u_{n}\right| ^{q_{1}}dx\leq \left( \left\| u_{n}\right\| ^{q_{1}}\right) \mathcal{S}_{0}\left( q_{1},r_{\varepsilon }\right) \leq \left( \sup_{n}\left\| u_{n}\right\|
^{q_{1}}\right) \mathcal{S}_{0}\left( q_{1},r_{\varepsilon }\right) <\frac{\varepsilon }{3}$$ and $$\int_{B_{R_{\varepsilon }}^{c}}K\left( \left| x\right| \right) \left|
u_{n}\right| ^{q_{2}}dx\leq \left( \sup_{n}\left\| u_{n}\right\| ^{q_{2}} \right) \mathcal{S}
_{\infty }\left( q_{2},R_{\varepsilon }\right) <\frac{\varepsilon }{3}.$$ Using (\[LEM:corone\]) of Lemma \[Lem(corone)\] and the boundedness of $\left\{ \left\|
u_{n}\right\| \right\} $ again, we infer that there exist two constants $
C_{2},l>0$, independent from $n$, such that $$\int_{B_{R_{\varepsilon }}\setminus B_{r_{\varepsilon }}}K\left( \left|
x\right| \right) \left| u_{n}\right| ^{q_{1}}dx\leq C_{2}\left(
\int_{B_{R_{\varepsilon }}\setminus B_{r_{\varepsilon }}}\left| u_{n}\right|
^{p}dx\right) ^{l},$$ where $$\int_{B_{R_{\varepsilon }}\setminus B_{r_{\varepsilon }}}\left| u_{n}\right|
^{p}dx\rightarrow 0\quad \text{as }n\rightarrow \infty \quad \text{(}\varepsilon ~\text{fixed)}$$ thanks to Lemma \[A6\]. Therefore we obtain $$\int_{B_{R_{\varepsilon }}}K\left( \left| x\right| \right) \left|
u_{n}\right| ^{q_{1}}dx+\int_{B_{R_{\varepsilon }}^{c}}K\left( \left|
x\right| \right) \left| u_{n}\right| ^{q_{2}}dx<\varepsilon$$ for all $n$ sufficiently large, which means $u_{n}\rightarrow 0$ in $L_{K}^{q_{1}}+L_{K}^{q_{2}}$ (Proposition \[Prop(-&gt;0)\]). This concludes the proof of part (ii).

Proof of Theorems \[THM0\] and \[THM1\] \[SEC:2\]
=================================================

Assume as usual $N \geq 3$ and $1<p<N$, and let $A$, $V$ and $K$ be as in $\left( \mathbf{A}\right)$, $\left( \mathbf{V}\right)$ and $\left( \mathbf{K}\right)$.

\[Lem(Omega)\]Let $R_0>0$ and assume that $V(r)<+\infty$ almost everywhere in $B_{R_0}$ and $$\Lambda :=\operatorname*{ess\,sup}_{x\in B_{R_0} }\frac{K\left( \left| x\right|
\right) }{\left| x\right| ^{\alpha }V\left( \left| x\right| \right) ^{\beta }}<+\infty \quad \text{for some }0\leq \beta \leq 1\text{~and }\alpha \in 
\mathbb{R}.$$ Let $u\in X$ and assume that there exist $\nu \in \mathbb{R}$ and $m>0$ such that $$\left| u\left( x\right) \right| \leq \frac{m}{\left| x\right| ^{\nu }}\quad 
\text{almost everywhere in } B_{R_0} .$$ Then there exists a constant $C=C(N, R_0, a_0, a_{\infty} ,\beta )>0$ such that $\forall R \in (0,R_{0})$ and $\forall q>\max \left\{ 1,p\beta \right\} $, one has

$\displaystyle\int_{B_R }K\left( \left| x\right| \right) \left| u\right| ^{q} dx$ $$\leq \left\{ 
\begin{array}{ll}
\Lambda m^{q-1} C \left( \int_{B_R}\left| x\right| ^{\frac{\alpha -\nu \left( q-1\right) }{N(p-1)+p\left( 1-p\beta +a_0 \beta \right) -a_0}pN}dx\right)^{
\frac{N(p-1)+p\left( 1-p\beta +a_0 \beta \right) -a_0}{pN}}\left\| u\right\| \quad \medskip  & 
\text{if }0\leq \beta \leq \frac{1}{p} \\ 
\Lambda m^{q-p\beta }\left( \int_{B_R }\left| x\right| ^{\frac{\alpha
-\nu \left( q-p\beta \right) }{1-\beta }}dx\right) ^{1-\beta }\left\|
u\right\| ^{p\beta } \medskip  & \text{if }\frac{1}{p}
<\beta <1 \\ 
\Lambda m^{q-p}\left( \int_{B_R }\left| x\right| ^{\frac{p}{p-1}(\alpha -\nu \left(
q-p\right)) }V\left( \left| x\right| \right) \left| u\right| ^{p}dx\right) ^{
\frac{p-1}{p}}\left\| u\right\|  & \text{if }\beta =1.
\end{array}
\right.$$

We distinguish several cases, where we will use Hölder inequality many times, without explicitly noting it.

*Case* $\beta =0$*.*

We apply Hölder inequality with exponents $p_0 = \frac{Np}{N-p +a_0}$, $(p_0 )' = \frac{Np}{N(p-1) -a_0 +p}$, and we use Lemma \[A4\]. We have [$$\begin{aligned}
\frac{1}{\Lambda }\int_{B_R }K\left( \left| x\right| \right) \left|
u\right| ^{q} dx 
&\leq & \int_{B_R }\left| x\right|^{\alpha }\left| u\right| ^{q-1}\left| u\right| dx \\
&\leq & \left( \int_{B_R
}\left( \left| x\right| ^{\alpha }\left| u\right| ^{q-1}\right) ^{\frac{pN}{
N(p-1)+p-a_0}}dx\right) ^{\frac{N(p-1)+p-a_0}{pN}}\left( \int_{B_R }\left| u\right|
^{p_0}dx\right) ^{\frac{1}{p_0}} \\
&\leq & m^{q-1} C \left( \int_{B_R }\left| x\right| ^{\frac{\alpha -\nu
\left( q-1\right) }{N(p-1)+p- a_0}pN}dx\right) ^{\frac{N(p-1)+p -a_0}{pN}}\left\| u\right\| .\end{aligned}$$ ]{}

*Case* $0<\beta <1/p$*.*

One has $\frac{1}{\beta }>1$ and $\frac{1-\beta }{1-p\beta }
p_0 >1 $, with Hölder conjugate exponents $\left( \frac{1}{\beta }
\right) ^{\prime }=\frac{1}{1-\beta }$ and $\left( \frac{1-\beta }{1-p\beta }
p_0\right) ^{\prime }=\frac{pN\left( 1-\beta \right) }{N(p-1)+p\left( 1-p\beta
+a_0 \beta\right) -a_0}$. Then we get [$$\begin{aligned}
&&\frac{1}{\Lambda }\int_{B_R }K\left( \left| x\right| \right) \left|
u\right| ^{q} dx 
\\
&\leq &\int_{B_R }\left| x\right| ^{\alpha }V\left( \left| x\right|
\right) ^{\beta }\left| u\right| ^{q} dx=\int_{B_R
}\left| x\right| ^{\alpha }\left| u\right| ^{q-1}\left| u\right| ^{1-p\beta
}V\left( \left| x\right| \right) ^{\beta }\left| u\right| ^{p\beta }dx 
\\
&\leq &\left( \int_{B_R }\left( \left| x\right| ^{\alpha }\left| u\right|
^{q-1}\left| u\right| ^{1-p\beta }\right) ^{\frac{1}{1-\beta }}dx\right)
^{1-\beta }\left( \int_{B_R }V\left( \left| x\right| \right) \left|
u\right| ^{p}dx\right) ^{\beta } 
\\
&\leq & \left( \left( \int_{B_R}\left( \left| x\right| ^{\frac{\alpha }{
1-\beta }}\left| u\right| ^{\frac{q-1}{1-\beta }}\right) ^{\left( \frac{
1-\beta }{1-p\beta }p_0\right) ^{\prime }}dx\right) ^{\frac{1}{\left( 
\frac{1-\beta }{1-p\beta }p_0\right) ^{\prime }}}\left( \int_{B_R
}\left| u\right| ^{p_0}dx\right) ^{\frac{1-p\beta }{\left( 1-\beta \right)
p_{0}}}\right) ^{1-\beta }\left\| u\right\| ^{p\beta } 
\\
&\leq &m^{q-1} C \left( \left( \int_{B_R}\left( \left| x\right| ^{\frac{
\alpha }{1-\beta }-\nu \frac{q-1}{1-\beta }}\right) ^{\left( \frac{1-\beta }{
1-p\beta }p_{0}\right) ^{\prime }}dx\right) ^{\frac{1}{\left( \frac{1-\beta 
}{1-\beta }p_{0}\right) ^{\prime }}} C
\left\| u\right\| ^{\frac{1-p\beta }{1-\beta }}\right)^{1-\beta }\left\|
u\right\| ^{p\beta } 
\\
&=&  m^{q-1} C \left( \int_{B_R }\left| x\right| ^{\frac{\alpha -\nu \left(
q-1\right) }{N(p-1)+p\left( 1-p\beta +a_0 \beta \right)-a_0 }pN}dx\right) ^{\frac{N(p-1)+p\left(
1-p\beta +a_0 \beta \right) -a_0}{pN}} C \left\| u\right\| .\end{aligned}$$ ]{}

*Case* $\beta =\frac{1}{p}$*.*

We have [$$\begin{aligned}
\frac{1}{\Lambda }\int_{B_R }K\left( \left| x\right| \right) \left|
u\right| ^{q} dx &\leq &\int_{B_R }\left| x\right|
^{\alpha }\left| u\right| ^{q-1}V\left( \left| x\right| \right) ^{\frac{1}{p}}\left| \right| dx \\u
&\leq &\left( \int_{B_R }\left| x\right| ^{\alpha \frac{p}{p-1}}\left| u\right|
^{\left( q-1\right)\frac{p}{p-1} }dx\right) ^{\frac{p-1}{p}}\left( \int_{B_R }V\left(
\left| x\right| \right) \left| u\right| ^{p}dx\right) ^{\frac{1}{p}} \\
&\leq &m^{q-1}\left( \int_{B_R}\left| x\right| ^{(\alpha -\nu \left(
q-1\right) )\frac{p}{p-1} }dx\right) ^{\frac{p-1}{p}}\left\| u\right\| .\end{aligned}$$ ]{}

*Case* $1/p<\beta <1$.

One has $\frac{p-1}{p\beta -1}>1$, with Hölder conjugate exponent $\left( \frac{p-1}{p\beta -1}\right) ^{\prime }=\frac{p-1}{p\left(
1-\beta \right) }$. Then [$$\begin{aligned}
&&\frac{1}{\Lambda }\int_{B_R }K\left( \left| x\right| \right) \left|
u\right| ^{q}dx 
\\
&\leq &\int_{B_R }\left| x\right|
^{\alpha }V\left( \left| x\right| \right) ^{\beta }\left| u\right|
^{q} dx=\int_{B_R }\left| x\right| ^{\alpha }V\left(
\left| x\right| \right) ^{\frac{p\beta -1}{p}}\left| u\right| ^{q-1}V\left(
\left| x\right| \right) ^{\frac{1}{p}}\left| u\right| dx 
\\
&\leq &\left( \int_{B_R}\left| x\right| ^{\alpha \frac{p}{p-1}}V\left( \left|
x\right| \right) ^{\frac{p\beta -1}{p-1}}\left| u\right| ^{\left( q-1\right) \frac{p}{p-1}}dx\right)
^{\frac{p-1}{p}}\left( \int_{B_R }V\left( \left| x\right| \right) \left|
u\right| ^{p}dx\right) ^{\frac{1}{p}} 
\\
&\leq &\left( \int_{B_R }\left| x\right| ^{\alpha \frac{p}{p-1}}\left| u\right|
^{(q-1)\frac{p}{p-1}- p\frac{p\beta -1}{p-1} }V\left( \left| x\right| \right) ^{\frac{p\beta -1}{p-1}}\left| u\right| ^{p \frac{p\beta -1}{p-1} }dx\right) ^{\frac{p-1}{p}}\left\| u\right\| 
\\
&\leq &\left( \left( \int_{B_R }\left| x\right| ^{\frac{\alpha }{1-\beta }}\left| u\right| ^{\frac{q-p\beta }{1-\beta }}dx\right) ^{\frac{p}{p-1}\left( 1-\beta
\right) }\left( \int_{B_R  }V\left( \left| x\right| \right) \left|
u\right| ^{p}dx\right) ^{\frac{p\beta -1}{p-1}}\right) ^{\frac{p-1}{p}}\left\| u\right\| 
\\
&\leq &m^{q-p\beta }\left( \int_{B_R  }\left| x\right|^{\frac{\alpha }{1-\beta }-\nu \frac{q-p\beta }{1-\beta }}
dx\right) ^{1-\beta }\left( \int_{B_R }V\left(
\left| x\right| \right) \left| u\right| ^{p}dx\right) ^{\frac{p\beta -1}{p}}\left\| u\right\| 
\\
&\leq &m^{q-p\beta }\left( \int_{B_R }\left| x\right| ^{\frac{\alpha -\nu
(q-p\beta )}{1-\beta }}dx\right) ^{1-\beta }\left\| u\right\| ^{p\beta
-1}\left\| u\right\| .\end{aligned}$$ ]{}

*Case* $\beta =1$*.*

Assumption $q>\max \left\{ 1,p\beta \right\} $ means $q>p$ and thus we have [$$\begin{aligned}
&&\frac{1}{\Lambda }\int_{B_R  }K\left( \left| x\right| \right) \left|
u\right| ^{q} dx 
\\
&\leq &\int_{B_R  }\left| x\right|
^{\alpha }V\left( \left| x\right| \right) \left| u\right| ^{q} dx 
= \int_{B_R  }\left| x\right| ^{\alpha }V\left( \left| x\right|
\right) ^{\frac{p-1}{p}}\left| u\right| ^{q-1}V\left( \left| x\right| \right)
^{\frac{1}{p}}\left| u\right| dx 
\\
&\leq &\left( \int_{B_R  }\left| x\right| ^{\alpha \frac{p}{p-1}}V\left( \left|
x\right| \right) \left| u\right| ^{\left( q-1\right)\frac{p}{p-1} }dx\right) ^{\frac{p-1}{p}}\left( \int_{B_R  }V\left( \left| x\right| \right) \left| u\right|
^{p}dx\right) ^{\frac{1}{p}} 
\\
&\leq &\left( \int_{B_R  }\left| x\right| ^{\alpha \frac{p}{p-1}}\left| u\right|
^{(q-1)\frac{p}{p-1}-p }V\left( \left| x\right| \right) \left| u\right|
^{p}dx\right) ^{\frac{p-1}{p}}\left\| u\right\| 
\\
&\leq &m^{q-p}\left( \int_{B_R }\left| x\right| ^{\frac{p}{p-1}\left(\alpha -\nu (q-p) \right) }V\left( \left| x\right| \right) \left| u\right| ^{p}dx\right) ^{\frac{p-1}{p}}\left\| u\right\| .\end{aligned}$$ ]{}

The following lemma is analogous to the previous one, so we skip its proof for brevity.

\[Lem(Omega2)\]Let $R_0>0$ and assume that $V(r)<+\infty$ almost everywhere in $B_{R_0^c}$ and $$\Lambda :=\operatorname*{ess\,sup}_{x\in B_{R_0^c} }\frac{K\left( \left| x\right|
\right) }{\left| x\right| ^{\alpha }V\left( \left| x\right| \right) ^{\beta }}<+\infty \quad \text{for some }0\leq \beta \leq 1\text{~and }\alpha \in 
\mathbb{R}.$$ Let $u\in X$ and assume that there exist $\nu \in \mathbb{R}$ and $m>0$ such that $$\left| u\left( x\right) \right| \leq \frac{m}{\left| x\right| ^{\nu }}\quad 
\text{almost everywhere on } B_{R_0^c} .$$ Then there exists a constant $C=C(N, R_0, a_0, a_{\infty} ,\beta )>0$ such that $\forall R > R_{0}$ and $\forall q>\max \left\{ 1,p\beta \right\} $, one has

$\displaystyle\int_{B_R^c }K\left( \left| x\right| \right) \left| u\right| ^{q-1}\left|
h\right| dx$ $$\leq \left\{ 
\begin{array}{ll}
\Lambda m^{q-1} C \left( \int_{B^c_R}\left| x\right| ^{\frac{\alpha -\nu \left( q-1\right) }{N(p-1)+p\left( 1-p\beta +a_{\infty} \beta \right) -a_{\infty}}pN}dx\right)^{
\frac{N(p-1)+p\left( 1-p\beta +a_{\infty} \beta \right)-a_{\infty} }{pN}}\left\| u\right\| \quad \medskip  & 
\text{if }0\leq \beta \leq \frac{1}{p} \\ 
\Lambda m^{q-p\beta }\left( \int_{B^c_R }\left| x\right| ^{\frac{\alpha
-\nu \left( q-p\beta \right) }{1-\beta }}dx\right) ^{1-\beta }\left\|
u\right\| ^{p\beta } \medskip  & \text{if }\frac{1}{p}
<\beta <1 \\ 
\Lambda m^{q-p}\left( \int_{B^c_R }\left| x\right| ^{\frac{p}{p-1}(\alpha -\nu \left(
q-p\right)) }V\left( \left| x\right| \right) \left| u\right| ^{p}dx\right) ^{
\frac{p-1}{p}}\left\| u\right\|  & \text{if }\beta =1.
\end{array}
\right.$$

We can now prove Theorems \[THM0\] and \[THM1\].

Assume the hypotheses of the theorem and let $u\in X$ be such that $\left\| u\right\| =1$. Let $0<R< R_{1}$. We will denote by $C$ any positive constant which does not depend on $u$ and $R$.

Recalling the pointwise estimates of Lemma \[lemdens2\] and the fact that $$\operatorname*{ess\,sup}_{x\in B_{R}}\frac{K\left( \left| x\right| \right) }{\left|
x\right| ^{\alpha _{0}}V\left( \left| x\right| \right) ^{\beta _{0}}}\leq 
\operatorname*{ess\,sup}_{r\in \left( 0,R_{1}\right) }\frac{K\left( r\right) }{r^{\alpha _{0}}V\left( r\right) ^{\beta _{0}}}<+\infty ,$$ we can apply Lemma \[Lem(Omega)\] with $R_0=R_1$, $\alpha =\alpha
_{0}$, $\beta =\beta _{0}$, $m=M\left\| u\right\| =M$ and $\nu =
\frac{N-p+a_0}{p}$.

If $0\leq \beta _{0}\leq 1/p$ we get

$$\int_{B_{R}}K\left( \left| x\right| \right) \left| u\right| ^{q_{1}} dx 
\leq C\left( \int_{B_{R}}\left| x\right|^{\frac{\alpha _{0}-
\nu\left( q_{1}-1\right) }{N(p-1)-a_0 +p\left( 1-p\beta _{0} +a_0 \beta_0 \right) }
pN}dx\right)^{\frac{N(p-1)-a_0 +p\left( 1-p\beta _{0} +a_0 \beta_0 \right) }{pN}}$$

$$\leq  C\left( \int_{0}^R r^{\frac{\alpha _{0}-
\nu\left( q_{1}-1\right) }{N(p-1)-a_0 +p\left( 1-p\beta _{0} +a_0 \beta_0 \right) }
pN  +N -1} dr\right)^{\frac{N(p-1)-a_0 +p\left( 1-p\beta _{0} +a_0 \beta_0 \right) }{pN}} .$$

Notice now that

$${\frac{\alpha _{0}-
\nu\left( q_{1}-1\right) }{N(p-1)-a_0 +p\left( 1-p\beta _{0} +a_0 \beta_0 \right) }
pN  +N }=$$

$$= 
\frac{N}{N(p-1) -a_0 +p (1-p\beta_0 +a_0 \beta_0)} \left[  p(N+ \alpha_0 -p \beta_0 + \alpha_0 \beta_0 )- (N+a_0 -p) q_1     \right] =$$

$$=\frac{N (N+a_0 -p)}{N(p-1) -a_0 +p (1-p\beta_0 +a_0 \beta_0)} \left[ q^* (a_0 , \alpha_0 , \beta_0 )-q_1 \right] >0,$$

thanks to the hypotheses. Hence we deduce

$$\int_{B_{R}}K\left( \left| x\right| \right) \left| u\right| ^{q_{1}} dx 
\leq C R^{\frac{N+a_0 -p}{p}\left[ q^* (a_0 , \alpha_0 , \beta_0 )-q_1 \right]}.$$

On the other hand, if $1/p<\beta _{0}<1$ we have $$\begin{aligned}
\int_{B_{R}}K\left( \left| x\right| \right) \left| u\right| ^{q_{1}} dx 
&\leq & C\left( \int_{B_{R}}\left| x\right| ^{\frac{\alpha _{0}-
\nu\left( q_{1}-p\beta _{0}\right) }{1-\beta _{0}}}dx\right)^{1-\beta _{0}}\\
&\leq & C\left( \int_{0}^{R}r^{\frac{\alpha _{0}-\nu
\left( q_{1}-p\beta _{0}\right) }{1-\beta _{0}}+N-1}dr\right) ^{1-\beta _{0}},\end{aligned}$$ where

$$\frac{\alpha_0 -\nu (q_1 - p \beta_0 )}{1-\beta_0} +N = \frac{p\alpha_0 -(N+\alpha_0 -p) q_1 -p\beta_0}{p(1-\beta_0 )} +N=$$

$$\frac{p\alpha_0 -(N+ a_0 -p) q_1 + Np\beta_0 + pa_0 \beta_0 -p^2 \beta_0 +Np -Np\beta_0 }{p(1-\beta_0 )}=$$

$$\frac{p(\alpha_0 -p \beta_0 +N + a_0 \beta_0 )- (N+a_0 - p)q_1}{p(1-\beta_0 )}=
\frac{N+a_0 - p }{p(1-\beta_0 )} \left[ q^* (a_0 , \alpha_0 , \beta_0 )-q_1 \right] >0 .$$

Hence we get

$$\int_{B_{R}}K\left( \left| x\right| \right) \left| u\right| ^{q_{1}} dx 
\leq C R^{\frac{N + a_0 -p}{p} \left[ q^* (a_0 , \alpha_0 , \beta_0 )-q_1 \right]}.$$

Finally, if $\beta _{0}=1$, we obtain

$$\int_{B_{R}} K\left( \left| x\right| \right) \left| u\right|^{q_{1}} dx \leq $$ C ( \_[B\_R]{}| x|\^[ (\_0 -( q\_1 -p )) ]{} V ( | x| ) |u|\^[p]{} dx)\^, $$\noindent where$$\_0 - (q\_1 -p) = \_0 -(q\_1 -p)= ( p \_0 -( N+a\_0 -p) ) q\_1 +pN +p a\_0 -p\^2 )=$$$$ ( p ( \_0 -p +N +a\_0 ) - ( N+a\_0 -p ) q\_1 )= ( q\^\* (a\_0, \_0 ,1 ) - q\_1 ) &gt;0.$$\noindent Hence we get$$ \_[B\_[R]{}]{} K( | x| ) | u|\^[q\_[1]{}]{} dx $$$$C( R\^[( q\^\* (a\_0, \_0 ,1 ) - q\_1 ) ]{}\_[B\_[R]{}]{}V( | x| ) | u| \^[p]{}dx) \^[ ]{} CR\^[ ( q\^\* (a\_0, \_0 ,1 ) - q\_1 ) ]{}. $$

So, in any case, we deduce $\mathcal{S}_{0}\left( q_{1},R\right) \leq
CR^{\delta }$ for some $\delta =\delta \left( N,p,\alpha _{0},\beta
_{0},q_{1}\right) >0$ and this concludes the proof.

Assume the hypotheses of the theorem and let $u \in X$ be such that $\left\| u\right\| =1$. Let $R> R_{2}$. We will denote by $C$ any positive constant which does not depend on $u$ and $R$.

By the pointwise estimates of Lemma \[lemdens2\] and the fact that $$\operatorname*{ess\,sup}_{x\in B_{R}^{c}}\frac{K\left( \left| x\right| \right) }{\left| x\right| ^{\alpha _{\infty }}V\left( \left| x\right| \right) ^{\beta
_{\infty }}}\leq \operatorname*{ess\,sup}_{r>R_{2}}\frac{K\left( r\right) }{r^{\alpha _{\infty }}V\left( r\right) ^{\beta _{\infty }}}<+\infty ,$$ we can apply Lemma \[Lem(Omega2)\] with $R_0= R_2$, $\alpha
=\alpha _{\infty }$, $\beta =\beta _{\infty }$, $m=M \left\|u\right\|
=M $ and $\nu =\frac{N-p+ a_{\infty}}{p}$.

If $0\leq \beta _{\infty }\leq 1/p$ we get

$$\begin{aligned}
\int_{B_{R}^{c}}K\left( \left| x\right| \right) \left| u\right|
^{q_{2}} dx &\leq &C\left( \int_{B_{R}^{c}}\left| x\right|
^{\frac{\alpha _{\infty }- \nu \left( q_{2}-1\right) }{N(p-1)-a_{\infty}+p\left(
1-p\beta _{\infty } +a_{\infty} \beta_{\infty}\right) }pN}dx\right) ^{\frac{N(p-1)-a_{\infty}+p\left( 1-p\beta _{\infty
}+a_{\infty} \beta_{\infty}\right) }{pN}}. \end{aligned}$$

Notice that we have

$$\frac{\alpha _{\infty }- \nu \left( q_{2}-1\right) }{N(p-1)-a_{\infty}+p\left(
1-p\beta _{\infty } +a_{\infty} \beta_{\infty}\right) } \, pN  +N =$$ $$= \frac{N \left( N+ a_{\infty} -p \right) }{N(p-1)-a_{\infty}+p\left(
1-p\beta _{\infty } +a_{\infty} \beta_{\infty}\right) }   \left[ q^* ( a_{\infty}, \alpha_{\infty}, \beta_{\infty}) -q_2 \right] <0 ,$$

thanks to the hypotheses. Hence we get

$$\int_{B_{R}^{c}}K\left( \left| x\right| \right) \left| u\right|
^{q_{2}} dx \leq$$ $$\leq C \left( \int_{R}^{+\infty} r^{\frac{N \left( N+ a_{\infty} -p \right) }{N(p-1)-a_{\infty}+p\left(
1-p\beta _{\infty } +a_{\infty} \beta_{\infty}\right) }   \left[ q^* ( a_{\infty}, \alpha_{\infty}, \beta_{\infty}) -q_2 \right]-1}dr \right)^{\frac{N(p-1)-
a_{\infty}+p\left( 1-p\beta _{\infty
}+a_{\infty} \beta_{\infty}\right) }{pN}}= C R^{\delta}$$

for some $\delta <0$. On the other hand, if $1/p<\beta _{\infty }<1$ we have $$\begin{aligned}
\int_{B_{R}^{c}}K\left( \left| x\right| \right) \left| u\right|
^{q_{2}} dx &\leq &C\left( \int_{B_{R}^{c}}\left| x\right|
^{\frac{\alpha _{\infty }-\nu \left( q_{2}-p\beta _{\infty }\right) 
}{1-\beta _{\infty }}}dx\right) ^{1-\beta _{\infty }} \\
&\leq &C\left( \int_{R}^{+\infty }r^{\frac{\alpha _{\infty }-\nu
\left( q_{2}-p\beta _{\infty }\right) }{1-\beta _{\infty }}+N-1}dr\right)
^{1-\beta _{\infty }},\end{aligned}$$

where

$$\frac{\alpha _{\infty }-\nu
\left( q_{2}-p\beta _{\infty }\right) }{1-\beta_{\infty }}+N=\frac{N+a_{\infty} -p}{p(1-\beta{\infty})}\left( q^* (a_{\infty},\alpha_{\infty}, \beta_{\infty})
-q_2 \right) <0.$$

Hence we get

$$\int_{B_{R}^{c}}K\left( \left| x\right| \right) \left| u\right|
^{q_{2}} dx \leq C R^{\frac{N+a_{\infty} -p}{p(1-\beta{\infty})}\left( q^* (a_{\infty},\alpha_{\infty}, \beta_{\infty})
-q_2 \right)}.$$

Finally, if $\beta _{\infty }=1$, we obtain $$\int_{B_{R}^{c}}K\left( \left| x\right| \right) \left| u\right|^{q_{2}} dx 
\leq C\left( \int_{B_{R}^{c}}\left| x\right|^{\frac{p}{p-1}(\left( \alpha _{\infty }-\nu \left( q_{2}-p\right) \right) }V\left( \left|x\right| \right) \left| u\right|^{p}dx\right)^{\frac{p-1}{p}} ,$$

where

$$\alpha_{\infty} -\nu (q_2 -p) = \frac{N+ a_{\infty} -p}{p} \left( q^* (a_{\infty}, \alpha_{\infty}, \beta_{\infty})-q_2 \right) <0 .$$

Hence

$$\int_{B_{R}^{c}}  K\left( \left| x\right| \right) \left| u\right|
^{q_{2}} dx \leq C R^{\frac{N+ a_{\infty} -p}{p} \left( q^* (a_{\infty}, \alpha_{\infty}, \beta_{\infty})-q_2 \right)}
\left(\int_{B_{R}^{c}} V(|x|) |u(x)|^p dx \right)^{\frac{p-1}{p}}\leq$$

$$\leq C R^{\frac{N+ a_{\infty} -p}{p} \left( q^* (a_{\infty}, \alpha_{\infty}, \beta_{\infty})-q_2 \right)}.$$

So, in any case, we get $\mathcal{S}_{\infty }\left(
q_{2},R\right) \leq CR^{\delta }$ for some $\delta =\delta ( N,p,\alpha_{\infty },\beta _{\infty },q_{2}) <0$, which completes the proof.

Existence of solutions \[SEC: ex\]
==================================

Let $N\geq 3$ and $1<p<N$. In this section we apply our embedding results to get existence of radial weak solutions to the equation $$-\mathrm{div}\left(A(|x| |\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N},  \label{EQg}$$ i.e., functions $u\in X$ such that $$\int_{\mathbb{R}^{N}} A(|x|) |\nabla u|^{p-2}\nabla u\cdot \nabla h\,dx+\int_{
\mathbb{R}^{N}}V\left( \left| x\right| \right) |u|^{p-2}uh\,dx=\int_{
\mathbb{R}^{N}}K(|x|) f(u) h\,dx\quad \forall h\in X,  \label{weak solution}$$ where $A$, $V$ and $K$ are potentials satisfying $\left( \mathbf{A}\right) $, $\left( \mathbf{V}\right) $ and $\left( \mathbf{K}\right) $, and $X$ and is the Banach spaces defined in Section 2.

We focus on super $p$-linear nonlinearities $f$ just for simplicity, but our compactness results also allow to treat the case of sub $p$-linear $f$’s. Moreover, multiplicity results can also be obtained. We leave the details to interested reader, which we refer to [@BGR_II; @BGR_p] for similar results and related arguments.

As concerns our hypotheses on the nonlinearity, we require that $f: {\mathbb{R}} \rightarrow {\mathbb{R}}$ is a continuous function, we set $F(t)= \int_{0}^{t} f(s)ds $, and we assume the following conditions:

-   $\exists \theta >p$ such that $0\leq \theta
    F\left( t\right) \leq f\left( t\right) t$ for all $t\in \mathbb{R}$;

-   $\exists t_{0}>0$ such that $F\left(t_{0}\right) >0$;

-   $\left|f\left( t\right) \right| \leq (\text{const.})\min \left\{ \left| t\right|
    ^{q_{1}-1},\left| t\right| ^{q_{2}-1}\right\} $ for all $t\in \mathbb{R}$.

We notice that these hypotheses imply $q_1, q_2 \geq \theta $. Also we observe that, if $q_{1}\neq q_{2}$, the double-power growth condition $\left( f_{q_{1},q_{2}}\right) $ is more stringent than the more usual single-power one, since it implies $|f(t)| \leq (\text{const.})|t|^{q-1}$ for $q=q_{1}$, $q=q_{2}$ and every $q$ in between. On the other hand, we will never require $q_{1}\neq q_{2}$ in $
\left( f_{q_{1},q_{2}}\right) $, so that our results will also concern single-power nonlinearities as long as we can take $q_{1}=q_{2}$ (cf. Example \[ex2\] below).

We set $$I\left( u\right) :=\frac{1}{p} \left\| u\right\| ^{p}-\int_{\mathbb{R}
^{N}}  K(|x|) F\left( u\right) dx=$$ $$=\frac{1}{p} \int_{\mathbb{R}^{N}} A(|x|) |\nabla u|^p dx + \frac{1}{p} \int_{\mathbb{R}^{N}} V(|x|) |u|^p dx - 
\int_{\mathbb{R}^{N}}  K(|x|) F\left( u\right) dx.$$

From the continuous embedding result of Theorem \[THM(cpt)\] and the results of [@BPR] about Nemytskiĭ operators on the sum of Lebesgue spaces, we have that $I$ is a $C^{1}$ functional on $X$ provided that there exist $q_{1},q_{2}>1$ such that $\left(
f_{q_{1},q_{2}}\right) $ and $\left( \mathcal{S}_{q_{1},q_{2}}^{\prime
}\right) $ hold. In this case, the Fréchet derivative of $I$ at any $
u\in X$ is given by $$I^{\prime }\left( u\right) h=\int_{\mathbb{R}^{N}}A(|x|)\left( |\nabla u|^{p-2}\nabla
u\cdot \nabla h+V\left( \left| x\right| \right) |u|^{p-2}uh\right) dx-\int_{
\mathbb{R}^{N}}K(|x|) f\left( u\right) h\,dx
\label{PROP:diff: I'(u)h=}$$ for all $h\in X$, and therefore the critical points of $I:X\rightarrow \mathbb{R}$ satisfy (\[weak solution\]).

Our existence result is the following.

\[THM:ex\]Assume that there exist $q_{1},q_{2}>p$ such that $\left( f_{q_{1},q_{2}}\right) $ and $\left( 
\mathcal{S}_{q_{1},q_{2}}^{\prime \prime }\right) $ hold. Then the functional $I:X\rightarrow \mathbb{R}$ has a nonnegative critical point $u\neq 0$.

\[RMK:thm:ex\] In Theorem \[THM:ex\], the assumptions on $f$ need only to hold for $t\geq
0$. Indeed, all the hypotheses of the theorem still hold true if we replace $
f\left( t\right) $ with $\chi _{\mathbb{R}_{+}}\left( t\right) f\left(
t\right) $ ($\chi _{\mathbb{R}_{+}}$ is the characteristic function of $\mathbb{R
}_{+}$) and this can be done without restriction since the theorem concerns nonnegative critical points.

The above result relies on assumption $\left( 
\mathcal{S}_{q_{1},q_{2}}^{\prime \prime }\right) $, which is quite abstract but can be granted in concrete cases through Theorems \[THM0\] and \[THM1\], which ensure such assumption for suitable ranges of exponents $q_{1}$ and $q_{2}$ by explicit conditions on the potentials. As concerns examples of nonlinearities satisfying the hypotheses of Theorem \[THM:ex\], the simplest $f\in C\left(\mathbb{R};\mathbb{R}\right) $ such that $\left( f_{q_{1},q_{2}}\right) $ holds is $$f\left( t\right) =\min \left\{ \left| t\right| ^{q_{1}-2}t,\left| t\right|
^{q_{2}-2}t\right\} ,$$ which also ensures $\left( f_{1}\right) $ if $
q_{1},q_{2}>p$ (with $\theta =\min \left\{ q_{1},q_{2}\right\} $). Another model example is $$f\left( t\right) =\frac{\left| t\right| ^{q_{2}-2}t}{1+\left| t\right|
^{q_{2}-q_{1}}}\quad \text{with }1<q_{1}\leq q_{2},$$ which ensures $\left( f_{1}\right) $ if $q_{1}>p$ (with $\theta =q_{1}$). Note that, in both these cases, also $\left( f_{2}\right) $ holds true. Moreover, both of these functions $f$ become $f\left( t\right) =\left| t\right| ^{q-2}t$ if $q_{1}=q_{2}=q$.

We now prove Theorem \[THM:ex\], starting with some lemmas.

\[LEM:MPgeom\]Assume the hypotheses of Theorem \[THM:ex\]. Then there exist three constants $c_{1},c_{2}>0$ such that $$I\left( u\right) \geq \frac{1}{p}\left\| u\right\| ^{p}-c_{1}\left\|
u\right\| ^{q_{1}}-c_{2}\left\| u\right\| ^{q_{2}}
\qquad \text{for all }u\in X.
  \label{LEM:MPgeom: th}$$

Let $i\in \left\{ 1,2\right\} $. By the monotonicity of $\mathcal{S}_{0}$ and $\mathcal{S}_{\infty }$, it is not restrictive to assume $R_{1}<R_{2}$ in hypothesis $\left( \mathcal{S}_{q_{1},q_{2}}^{\prime }\right) $. Then, by lemmas \[Lem(corone)\] and \[A6\], there exists a constant $c_{R_{1},R_{2}} >0$ such that for all $u\in X$ we have

$$\int_{B_{R_{2}}\setminus B_{R_{1}}}K\left( \left| x\right| \right) \left|
u\right| ^{q_{1}}dx\leq C_{R_{1},R_{2}}\left\| u\right\|
^{q_{1}}.$$

Therefore, by the hypotheses on $f$ and the definitions of $\mathcal{S}_{0}$ and $
\mathcal{S}_{\infty }$, we obtain $$\begin{aligned}
&&\left| \int_{\mathbb{R}^{N}}K\left( \left| x\right| \right) F\left( u\right) dx\right| \leq
\nonumber
\\
&\leq & C \int_{\mathbb{R}^{N}}K\left( \left| x\right| \right) \min
\left\{ \left| u\right| ^{q_{1}},\left| u\right| ^{q_{2}}\right\}
dx \nonumber \\
&\leq & C \left( \int_{B_{R_{1}}}K\left( \left| x\right| \right)
\left| u\right| ^{q_{1}}dx+\int_{B_{R_{2}}^{c}}K\left( \left| x\right|
\right) \left| u\right| ^{q_{2}}dx+\int_{B_{R_{2}}\setminus
B_{R_{1}}}K\left( \left| x\right| \right) \left| u\right| ^{q_{1}}dx\right)
 \nonumber \\
&\leq & C \left( \left\| u\right\| ^{q_{1}}\mathcal{S}_{0}\left(
q_{1},R_{1}\right) +\left\| u\right\| ^{q_{2}}\mathcal{S}_{\infty }\left(
q_{2},R_{2}\right) +C_{R_{1},R_{2}}\left\| u\right\|
^{q_{1}}\right)   
\\
&=&C_{1}\left\| u\right\| ^{q_{1}}+C_{2}\left\| u\right\|
^{q_{2}},  \nonumber\end{aligned}$$ where the constants $c_{1}$ and $c_{2}$ are independent of $u$. This yields (\[LEM:MPgeom: th\]).

\[LEM:PS\]Under the assumptions of Theorem \[THM:ex\], the functional $
I:X\rightarrow \mathbb{R}$ satisfies the Palais-Smale condition.

Let $\left\{ u_{n}\right\} $ be a sequence in $X$ such that $\left\{
I\left( u_{n}\right) \right\} $ is bounded and $I^{\prime }\left(
u_{n}\right) \rightarrow 0$ in $X^{\prime }$. Hence $$\frac{1}{p}\left\| u_{n}\right\| ^{p}-\int_{\mathbb{R}^{N}}K(|x|) F\left( u_{n}\right) dx=O\left( 1\right) \quad \text{and}\quad \left\|
u_{n}\right\| ^{p}-\int_{\mathbb{R}^{N}}K(|x|) f\left( u_{n}\right)
u_{n}dx=o\left( 1\right) \left\| u_{n}\right\| .$$ As $f$ satisfies $\left( f_{1}\right) $, we get $$\frac{1}{p}\left\| u_{n}\right\| ^{p}+O\left( 1\right) =\int_{\mathbb{R} 
^{N}} K(|x|)F\left( u_{n}\right)  dx\leq \frac{1}{\theta }\int_{\mathbb{R}^{N}}K(|x|)f\left( u_{n}\right) u_{n}dx=\frac{1}{\theta }\left\| u_{n}\right\| ^{p}+o\left( 1\right) \left\| u_{n}\right\| ,$$ which implies that $\left\{ \left\| u_{n}\right\| \right\} $ is bounded since $\theta >p$. Now, thanks to assumption $\left( \mathcal{S}_{q_{1},q_{2}}^{\prime \prime
}\right) $, we apply Theorem \[THM(cpt)\] to deduce the existence of $u\in
X$ such that (up to a subsequence) $u_{n}\rightharpoonup u$ in $X$ and $u_{n}\rightarrow u$ in $L_{K}^{q_{1}}+L_{K}^{q_{2}}$. Setting $$I_{1}\left( u\right) :=\frac{1}{p}\left\| u\right\| ^{p}\quad \text{and}
\quad I_{2}\left( u\right) :=I_{1}\left( u\right) -I\left( u\right)$$ for brevity, we have that $I_{2}$ is of class $C^{1}$ on $L_{K}^{q_{1}}+L_{K}^{q_{2}}$ by [@BPR Proposition 3.8] and therefore we get $\left\| u_{n}\right\| ^{p}=I^{\prime }\left( u_{n}\right)
u_{n}+I_{2}^{\prime }\left( u_{n}\right) u_{n}=I_{2}^{\prime }\left(
u\right) u+o\left( 1\right) $. Hence $\lim_{n\rightarrow \infty }\left\|
u_{n}\right\| $ exists and one has $\left\| u\right\| ^{p}\leq
\lim_{n\rightarrow \infty }\left\| u_{n}\right\| ^{p}$ by weak lower semicontinuity. Moreover, the convexity of $I_{1}:X\rightarrow \mathbb{R}$ implies $$I_{1}\left( u\right) -I_{1}\left( u_{n}\right) \geq I_{1}^{\prime }\left(
u_{n}\right) \left( u-u_{n}\right) =I^{\prime }\left( u_{n}\right) \left(
u-u_{n}\right) +I_{2}^{\prime }\left( u_{n}\right) \left( u-u_{n}\right)
=o\left( 1\right)$$ and thus $$\frac{1}{p}\left\| u\right\| ^{p}=I_{1}\left( u\right) \geq
\lim_{n\rightarrow \infty }I_{1}\left( u_{n}\right) =\frac{1}{p}\lim_{n\rightarrow \infty }\left\| u_{n}\right\| ^{p}.$$ So $\left\| u_{n}\right\| \rightarrow \left\| u\right\| $ and one concludes that $u_{n}\rightarrow u$ in $X$ by the uniform convexity of the norm.

Assume the hypotheses of the theorem, together with the following non-restrictive additional condition: $f(t)=0$ for $t<0$. We want to apply the Mountain-Pass Theorem. To this end, from (\[LEM:MPgeom: th\]) of Lemma \[LEM:MPgeom\] we deduce that, since $q_{1},q_{2}>p$, there exists $\rho >0$ such that $$\inf_{u\in X,\,\left\| u\right\| =\rho }I\left( u\right) >0=I\left(
0\right) .  \label{mp-geom}$$ Therefore, taking into account Lemma \[LEM:PS\], we need only to check that $\exists \bar{u}\in X$ such that $\left\| \bar{u}\right\| >\rho $ and $I\left( \bar{u}\right) <0$. To this end, from assumption $\left( f_{1}\right) $ and $
\left( f_{2}\right) $ we infer that $$F\left( t\right) \geq \frac{F\left( t_{0}\right) }{t_{0}^{\theta }}
t^{\theta }\text{ for all }t\geq t_{0}.$$ We then fix a non negative function $u_{0}\in C_{c}^{\infty }( \mathbb{R}^{N} \backslash \{0\}  )$ such that the set $\{x\in \mathbb{R}^{N}:u_{0}\left(
x\right) \geq t_{0}\}$ has positive Lebesgue measure. Hence, since $\left( f_{1}\right) $ and $\left( f_{2}\right) $ ensure that $F(t)\geq 0$ for all $t$ and $F\left( t_{0}\right) >0$, for every $\lambda >1$ we get $$\begin{aligned}
\int_{\mathbb{R}^{N}}K(|x|) F\left( \lambda u_{0}\right) dx &\geq
&\int_{\left\{ \lambda u_{0}\geq t_{0}\right\} }K(|x|) F\left(\lambda u_{0}\right) dx\geq \frac{\lambda ^{\theta }}{t_{0}^{\theta }}
\int_{\left\{ \lambda u_{0}\geq t_{0}\right\} }K(|x|) F\left( t_{0}\right) u_{0}^{\theta }dx \\
&\geq &\frac{\lambda ^{\theta }}{t_{0}^{\theta }}\int_{\left\{ u_{0}\geq
t_{0}\right\} }K(|x|) F\left( t_{0}\right) u_{0}^{\theta }dx\geq
\lambda ^{\theta }\int_{\left\{ u_{0}\geq t_{0}\right\} }K(|x|) F\left( t_{0}\right) dx>0.\end{aligned}$$ Since $\theta >p$, this gives $$\lim_{\lambda \rightarrow +\infty }I\left( \lambda u_{0}\right) \leq
\lim_{\lambda \rightarrow +\infty }\left( \frac{\lambda ^{p}}{p}\left\|
u_{0}\right\| ^{p}-\lambda ^{\theta }\int_{\left\{ u_{0}\geq t_{0}\right\}
} K(|x|) F\left( t_{0}\right) dx\right) =-\infty .$$ As a conclusion, we can take $\bar{u}=\lambda u_{0}$ with $\lambda $ sufficiently large and the Mountain-Pass Theorem provides the existence of a nonzero critical point $u\in X$ for $I$. Since the additional assumption $f(t)=0$ for $t<0$ implies $
I^{\prime }\left( u\right) u_{-}=-\left\| u_{-}\right\| ^{p}$ (where $
u_{-}\in X$ is the negative part of $u$), one concludes that $u_{-}=0$, i.e., $u$ is nonnegative.

Examples {#SEC:EX}
========

In this section we give some examples that might help to understand what is new in our results. We will make a comparison, in concrete cases, between our results and those of [@Su-Wang]. In that paper the authors prove some compactness theorems which are used to prove existence results for equation (\[EQ\]), where $f$ is a power or a sum of powers. We will show some cases where the results of [@Su-Wang] do not apply, while our results give existence of solutions. In all our example we look for a nonlinearity defined by $f(t)= \min \{ t^{q_1 -1} , t^{q_2 -1} \}$, and we will see how to choose $p<q_1 \leq q_2$ in such a way to get existence results for problem (\[EQ\]).

\[ex1\] Let $A, V, K$ be as follows:

$$A(r)= \min \{ r^{1/2} , r^{3/2} \}, \quad V(r)= \min \left\{ 1, r^{-3/2}  \right\}, \quad K(r)= \max \{ r^{1/2}, r^{3/2}  \}.$$

Assume $3/2 <p \leq 2$. We first show that in this case the results of [@Su-Wang] do not apply. The embedding theorems of that paper are Theorems 3.2, 3.3 and 3.4. If we compute the coefficients $q^*$ and $q_*$ of [@Su-Wang], we easily obtain $ q_{*}= \frac{p(2N+3)}{2N+1-2p}$, $ q^{*}= \frac{p(2N+1)}{2N+3-2p}$ and this, together with $p\leq 2$, implies $q^* < q_*$, so that Theorem 3.2 of [@Su-Wang] cannot be applied, because it needs $ q_* < q^*$. One easily verifies that also the hypotheses of Theorems 3.3 and 3.4 of [@Su-Wang] are not satisfied. To apply our results, we set $\beta_0 = \beta_{\infty}=0$, $\alpha_0 = 1/2$, $\alpha_{\infty}=3/2$, $a_0= 3/2$, $a_{\infty}=1/2$. Note that condition $a_0 , a_{\infty} \in (p-N,p]$ is satisfied. We apply Theorems \[THM0\] and \[THM1\], and we compute

$$q^{*} (a_0 , \alpha_0, \beta_0 ) = \frac{p(2N+1)}{2N+3-2p}, \quad  q_{*} (a_{\infty} , \alpha_{\infty}, \beta_{\infty} )=\frac{p(2N+3)}{2N+1-2p}.$$

Notice that these are the same value obtained above, following [@Su-Wang]. Notice also that $q^{*} (a_0 , \alpha_0, \beta_0 ) >p$ is equivalent to $p> 1 $. Applying our results, we deduce that if we take $q_1 ,q_2$ such that

$$p<q_1 < \frac{p(2N+1)}{2N+3-2p}< \frac{p(2N+3)}{2N+1-2p}<q_2 ,$$

then $\left(\mathcal{S}_{q_{1},q_{2}}^{\prime \prime }\right) $ holds and we get an existence result for the equation (\[EQ\]) with any nonlinearity satisfying $\left( f_{q_{1},q_{2}}\right) $ .

\[ex2\] Assume $N\geq 4$ and choose the functions $A, V, K$ as follows:

$$A(r)=  \max \left\{ r^{-2}, r^{-1} \right\} , \quad V(r)= e^{2r} , \quad K(r)= e^r  .$$

In this case the results of [@Su-Wang] do not apply because of the exponential growth of the potential $K$. Assume $1<p<N-2$. In order to apply Theorems \[THM0\] and \[THM1\], we can choose $a_0= -2 $, $a_{\infty} =-1$, $\beta_0 =\alpha_{0}= \alpha_{\infty}=0$, $\beta_{\infty}=1/2$. Notice that condition $a_0 , a_{\infty} \in (p-N,p]$ is satisfied. We get $$q^* (a_0, \alpha_0 , \beta_0 ) = \frac{pN}{N-p-2}\quad \text{and}\quad 
q^* (a_{\infty}, \alpha_{\infty} , \beta_{\infty} ) =\frac{pN}{N-p-1},$$ where we have $
p<\frac{pN}{N-p-1}<\frac{pN}{N-p-2}.
$ Then $\left(\mathcal{S}_{q_{1},q_{2}}^{\prime \prime }\right) $ holds and we get an existence result for the equation (\[EQ\]) with any nonlinearity satisfying $\left( f_{q_{1},q_{2}}\right) $ provided that $$p<q_1 < \frac{pN}{N-p-2}\quad \text{and}\quad  q_2 > \frac{pN}{N-p-1}.$$ In particular we can take a power nonlinearity $f(t)= t^{q-1}$ for $q \in \left( \frac{pN}{N-p-1}, \frac{pN}{N-p-2} \right)$.

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[^1]: Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy. e-mails: `<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>`, `<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>`

[^2]: Partially supported by the PRIN2012 grant “Aspetti variazionali e perturbativi nei problemi differenziali nonlineari”.

[^3]: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Roberto Cozzi 53, 20125 Milano, Italy. e-mail: `<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>`