# Existence of Solutions for p-Kirchhoff Problem of Brézis-Nirenberg Type with Singular Terms

Authors: Atika Matallah; Hayat Benchira; M. El Mokhtar Ould El Mokhtar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we prove the existence of positive solution for a p-Kirchhoff problem of Brézis-Nirenberg type with singular terms, nonlocal term, and the Caffarelli-Kohn-Nirenberg exponent by using variational methods, concentration compactness, and maximum principle.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the following p-Kirchhoff problem of Brézis-Nirenberg type with singular terms(1)−Kudiv∇up−2∇uxpα+μup−2uxpα+1=up∗−2uxβp∗+λup−2uxγ,inΩ,u=0,on∂Ω,where Ω is a bounded smooth domain in ℝN, 1<p<N,0∈Ω, Ku=auq+b,a,b,q>0,0≤α<N−p/p,α≤β<α+1,0≤γ<p,0≤μ<μ¯≔N−α+1p/pp,λ>0, p∗=pN/N−p1+α−β is the critical Caffarelli-Kohn-Nirenberg exponent corresponding to the noncompact embedding of DαΩ into Lp∗Ω,xβp∗, where DαΩ is the closure of C0∞Ω with respect to the norm
(2)up≔∫Ω∇upxpα−μupxpα+1dx,and Lp∗Ω,xβp∗ denotes the usual weighted Lp∗Ω space with the weight xβp∗.Kirchhoff-type problems are often referred to as being nonlocal because of the presence of the termKu which implies that the equation in (1) is no longer a pointwise identity. In the case p=2 and α=β=γ=μ=0, it is analogous to the stationary version of equations that arise in the study of string or membrane vibrations, namely,
(3)utt−KuΔu=gx,uwhere u denotes the displacement and gx,u is the external force. Equations of this type were first proposed by Kirchhoff in 1883 [1] to describe the transversal oscillations of a stretched string.These problems serve also to model other physical phenomena as biological systems whereu describes a process which depends on the average of itself (for example, population density).In recent years, Kirchhoff-type problems received much attention, mainly after the famous article of Lions [2]; they have been studied in many papers by using variational methods, see [3–9] and the references therein.The problem (1) without nonlocal term a=0 and without singular terms α=β=γ=μ=0 has been treated by Brézis and Nirenberg [10] for p=2. Subsequently, an increasing number of researchers have paid attention to semilinear or quasilinear elliptic equations with critical exponent of Sobolev or Caffarelli-Kohn-Nirenberg, for example, see [11, 12] and the references therein.In [7], Naimen generalized the results of [10] to the nonlocal problem (1) with N=3 and without singular terms. Kang in [1] generalized the results of [10] to a quasilinear problem with singular terms and without the nonlocal term p>1,a=0andα,β,γ,μ≠0,0,0,0.Thus, it is natural for us to consider the quasilinear Brézis-Nirenberg problem in [10] with nonlocal term and singular weights, p>1,a≠0andα,β,γ,μ≠0,0,0,0. The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of DαΩ into Lp∗Ω,xβp∗ prevent us from using the variational methods in a standard way. So, motivated by all the works mentioned above, we prove existence results of our problem for large range of N and under some little, as possible, conditions on q. We show that the existence of solutions depends on the parameter λ and the position of q with respect to p∗−p. Here, we need more delicate estimates.To the best of our knowledge, many of the results are new for,q>1, and even in the case α,β,γ,μ=0,0,0,0.Our technique is based on variational methods and concentration compactness argument [2].Note that if=λ=0,b=1, and Ω=ℝN, the problem (1) has positive radial ground state solution uεε>0. Moreover, for any ε>0 the extremal functions
(4)vεx=ε−N−pp−αuεxεis a minimizer for
(5)S≔infu∈DαΩ\0up∫ℝNup∗/xβp∗dxp/p∗and satisfies
(6)vεp=∫ℝNvεp∗xβp∗dx=Sp∗p∗−p.On the other hand, for0≤α<N−p/p,α≤β<α+1,0≤γ<p, and 0≤μ<μ¯, we defined the first eigenvalue of the problem
(7)−div∇up−2xpα∇u−μup−2xpα+1u=λup−2uxγ,inΩu=0,on∂Ωas
(8)λ1≔infu∈DαΩ\0up∫Ωup/xγdx,and it is positive, (see for instance [1]).The main result is concluded as the following theorem.Theorem 1.
Let1<p<N, 0≤α<N−p/p2,α≤β<α+1,p2α+1−N/p−1≤γ<p,b>0,0<λ<bλ1, and 0≤μ<μ∗ with
(9)μ∗=N−p2α+1+γ1−ppN−γpp−1.
Then, the problem (1) has a positive solution in the following cases:
(1)
q=p∗−p and 0<a<S−p+q/p(2)
q<p∗−p and a>0Remark 2.
In the case whereλ=0 and Ω is a star-shaped domain with respect to the origin, we can easily verify that the problem (1) has no nontrivial solution by using a Pohozaev-type identity.This paper is organized as follows. In Section2, we study the variational framework and give some preliminary results. In Section 3, we show the existence result and we will prove Theorem 1.


## 2. Variational Framework and Preliminary Results

The starting point of the variational approach to problem (1) is the following Caffarelli-Kohn-Nirenberg inequality in [13] which is also called the Hardy-Sobolev inequality. Assume that 1<p<N,0≤α<N−p/p and α≤β<α+1, and then,
(10)∫Ωup∗xp∗βdx1/p∗≤C~∫Ω∇upxpαdx1pforallu∈C0∞Ω,for some positive constant C~. In the case where β=α+1, we have p∗=p,C~=1/μ¯ and we have the following Hardy inequality:
(11)∫Ωupxpα+1dx≤1μ¯∫Ω∇upxpαdx,forallu∈C0∞Ω.Definition 3.
We say thatu∈DαΩ\0 is a weak solution of equation 1 if
(12)Ku∫Ω∇up−2∇u∇vxpα−μup−2uvxpα+1dx−∫Ωup∗−2uxp∗β+λup−2uxγvdx=0for any v∈DαΩ.Next, we define the energy functional(13)Iλu=aq+puq+p+bpup−1p∗∫Ωup∗xp∗βdx−λp∫Ωupxγdx,associated to problem (1), for all u∈DαΩ.Notice that the functionalIλ is well defined in DαΩ and belongs to C1DαΩ,ℝ and a critical point of Iλ is a weak solution of problem (1).Definition 4.
Letc∈ℝ, a sequence un⊂DαΩ is called a PSc sequence (Palais-Smale sequence at level c) if
(14)Iλun⟶candIλ′un⟶0asn⟶+∞.
Letc∈ℝ. We say that Iλ satisfies the Palais-Smale condition at level c, if any PSc sequence contains a convergent subsequence in DαΩ.Lemma 5.
Assume1<p<N, 0≤α<N−p/p,α≤β<α+1,0≤γ<p,a,b>0,0<λ<bλ1, and q≤p∗−p.Letc∈ℝ+ and un⊂DαΩ be a PSc sequence for Iλ. Then,
(15)un⇀uinDαΩfor some u∈DαΩ with Iλ′u=0.Proof.
We have(16)Iλun⟶c,Iλ′un⟶0.That is,
(17)c+on1=Iλun,on1v=Iλ′un,v,for any v∈DαΩ.Then, asn⟶∞, it follows that
(18)c+on1−1p∗on1un=Iλun−1p∗Iλ′un,un≥ap∗−q+pq+pp∗unq+p+b−λλ1p∗−ppp∗unp.Asλ<bλ1 and q≤p∗−p, we obtain that un is bounded in DαΩ. Up to a subsequence if necessary, there exists a function u∈DαΩ such that un⇀u in DαΩ,un⇀u in Lp∗Ω,xp∗β,un⟶u in LrΩ,xp∗β, for all r<p∗ and un⟶u a.e on Ω. Then,
(19)Iλ′un,v=0forallv∈C0∞Ω,and thus Iλ′u=0. This completes the proof of Lemma 5.The following lemma is very important for giving the local Palais-Smale condition.Lemma 6.
Leta,b,q>0,σ≥1, and y~=a/σSq+p/p1/σ−1 for σ>1. For y≥0, we consider the function f:ℝ+⟶ℝ∗, given by
(20)fy=S−1yσ−aSqpy−b.
Then,(1)
Ifσ=1 and 0<a<S−q+p/p, then the equation fy=0 has a unique positive solution(21)y1=bSq/pS−q+p/p−aand fy≥0 for all y≥y1.(2)
Ifσ>1, then the equation fy=0 has a unique positive solution y2>y~ and fy≥0 for all y≥y2.Proof.
(1)
Forσ=1 and 0<a<S−q+p/p, we have(22)fy=SqpS−q+pp−ay−bthat is, the equation fy=0 has a unique positive solution
(23)y1=bS−q+p/p−aSq/p,and fy≥0 for all y≥y1.(2)
Forσ>1, we have f′y=σS−1yσ−1−aSq/p and(24)f′′y=σσ−1S−1yσ−2>0,∀y>0.
Then,f′y~=0,f′y<0 for y<y~ and f′y>0 for y>y~. Hence, f is concave function and
(25)miny≥0fy=fy~=−σ−1S−1aσSq+ppσσ−1<0.
Moreover, we havefy~<0 and limy⟶+∞fy=+∞; thus, from (25) and the concavity of f, we can conclude that the equation fy=0 has a unique positive solution y2>y~ and fy≥0 for all y≥y2.
Now, we prove an important lemma which ensures the local compactness of the Palais-Smale sequence forIλ.Fori=1,2¯, let yi be defined in Lemma 6 and define
(26)y∗=y1ifq=p∗−pand0<a<S−q+pp,y2ifq<p∗−panda>0.(27)Ayi≔ap∗−q+pq+pp∗Syipqq+pp+bp∗−ppp∗Syipq,and C∗≔Ay∗.Lemma 7.
Let1<p<N, 0≤α<N−p/p,α≤β<α+1,0≤γ<p,b>0, and 0<λ<bλ1. Assume that q=p∗−p and 0<a<S−q+p/p or q<p∗−p and a>0. Then, the functional Iλ satisfies PSc condition for all c<C∗.Proof.
Letun⊂DαΩ is a PSc sequence for Iλ with c<C∗. By the proof of Lemma 5, we have un is a bounded sequence in DαΩ. Hence, by the concentration compactness principle due to Lions [2], there exists a subsequence, still denoted by un, such that
(28)∇unpxpα−μunpxpα+1⇀dη≥∇upxpα−μupxpα+1+∑i∈Iηiδ~iunp∗xβp∗⇀dθ=up∗xβp∗+∑i∈Iθiδ~i,where I is an at most countable index set and δ~i is the Dirac mass at xi. Moreover, by the Sobolev-Hardy inequality we infer that
(29)ηi≥Sθip/p∗foralli∈I.
We claim thatI is finite and for any i∈I, for ε>0, let ϕε,ix be a smooth cut-off function centered at xi such that 0≤ϕε,ix≤1, and
(30)ϕε,ix=1inBxi,ε,0inΩ\Bxi,2ε,.∇ϕε,ix≤2ε.
Sinceϕε,iun is bounded in W01,pΩ and Iλ′un⟶0 as n⟶∞, it holds by Hölder’s inequality
(31)0=limε⟶0limn⟶∞Iλ′un,ϕε,jun=limε⟶0limn⟶∞Kun∫Ω∇unp−2xpα∇un∇ϕε,jun−μunp−2xpα+1unϕε,jundx−∫Ωunp∗−2xβp∗unϕε,jundx−λ∫Ωunq−1xγϕε,jundx≥b+aηiqpηi−θi.
Then,θi≥bηi+aηiq+p/p. Therefore, by (29), we deduce that
(32)θi=0orS−1θip∗−pp∗−aSqpθiqp∗−b≥0.
Assume by contradiction that there existsi0∈I such that θi0≠0. Set y=θi0q/p∗ and σ=p∗−p/q, then by (32) we get
(33)S−1yσ−aSqpy−b≥0.
It is clear thatσ≥1 thanks to q≤p∗−p. So, from (33) and the definition of f in Lemma 6 we get
(34)fy=S−1yσ−aSqpy−b≥0.We will discuss it in two cases:Case 1.
q=p∗−p,b>0 and 0<a<S−q+p/p.
According to Lemma6, we have fy1=0 and fy≥0 if y≥y1 with
(35)y1=bS−q+p/p−aSq/p,which implies that
(36)Sθi0pp∗≥Sy1pq≕B1.Case 2.
q<p∗−p,b>0, and a>0. In this case, from Lemma 6, we get fy2=0 and fy≥0 if y≥y2 with
(37)y2>aqp∗−pSq+ppqp∗−q+p,which implies that
(38)Sθi0pp∗≥Sy2pq≕B2.Hence, using (29), we deduce ηi0≥Sθi0p/p∗≥B1ifq=p∗−pand0<a<S−q+p/p,B2ifq<p∗−panda>0.By Young inequality we have(39)c=limn⟶∞Iλun−1q+pIλ′un,un=limn⟶∞qq+ppbunp+p∗−q+pq+pp∗∫Ωunp∗xp∗β−λqq+pp∫Ωunpxγdx≥qq+ppbup+ηi0+p∗−q+pq+pp∗∫Ωup∗xp∗β+θi0+−λqq+pp∫Ωupxγdx≥qq+ppb−λλ1up+p∗−q+pq+pp∗∫Ωup∗xp∗β+qq+ppbηi0++p∗−q+pq+pp∗θi0,we observe that q/q+ppb−λ/λ1>0,p∗−q−p≥0; thus, for j∈1,2 we get
(40)c≥qq+ppbηi0+p∗−q+pq+pp∗θi0≥1p−1q+pbBj+p∗−q+pq+pp∗Bjp∗pS−p∗p≥1p−1q+pbBj+p∗−q+pq+pp∗Bjp∗pS−p∗p+p∗−q+pq+pp∗aBjq+pp−p∗−q+pq+pp∗aBjq+pp+1p∗bBj−1p∗bBj≥p∗−q+pq+pp∗aBjq+pp+1p−1p∗bBj−p∗−q+pq+pp∗bBj+p∗−q+pq+pp∗Bjp∗pS−p∗p−p∗−q+pq+pp∗aBjq+pp≥p∗−q+pq+pp∗aBjq+pp+1p−1p∗bBj+p∗−q+pq+pp∗BjBjp∗−ppS−p∗p−aBjqp−b≥p∗−q+pq+pp∗aSyjpqq+pp+1p−1p∗bSyjpq+p∗−q+pq+pp∗SyjpqS−1yjp∗−pq−aSqpyj−b≥C∗+p∗−q+pq+pp∗Syjpqfyj≥C∗,since fyj=0 for j∈1,2 and C∗ defined in Lemma 7. Contradiction with c<C∗. Then, I is empty, which implies that
(41)∫Ωunp∗xp∗βdx⟶∫Ωup∗xp∗βdx.Now, setl=limun as n⟶+∞; then, we have
(42)Iλ′un,un=aunq+bunp−∫Ωunp∗xp∗βdx−λ∫Ωunpxγdx=on1(43)Iλ′un,v=aunq+b∫Ω∇unp−2xpα∇un∇v−μunp−2xpα+1unvdx−∫Ωunp∗−2xp∗βunvdx−λ∫Ωunp−2xγunvdx=on1for any v∈DαΩ. Let n⟶+∞, and then, from (42) and (43), we deduce that
(44)alq+blp−∫Ωup∗xp∗βdx−λ∫Ωupxγdx=0,(45)alq+b∫Ω∇up−2xpα∇u∇v−μup−2uvxpα+1dx−∫Ωup∗−2uvxp∗βdx−λ∫Ωup−2uvxγdx=0.Taking the test functionv=u in (45), we get
(46)alq+bup−∫Ωup∗xp∗βdx−λ∫Ωupxγdx=0.Therefore, the equalities (44) and (45) imply that u=l. Consequently, un converges strongly in DαΩ, which is the desired result.


## 3. Proof of the Main Result

LetR be a positive constant and set φ∈C0∞Ω such that 0≤φx≤1 for x≤R and φx≡1 for x≤R/2 and BR0⊂Ω. Set zεx=φxuεx.We have the well-known estimates asε⟶0:
(47)zεp=Sp∗p∗−p+Oεpωμ+pα+1−N∫Ωzεp∗xβp∗dx=Sp∗p∗−p+Oεωμ+βp∗−N∫Ωzεpxγdx≥Cεpα+1−γifγ>N−pωμand0≤α<N−p/p2Cεpωμ−N+pα+1lnεifγ=N−pωμand0≤α<N−pp2,where ξμ and ωμ are zeroes of the function
(48)ft=p−1tp−N−pα+1tp−1+μ,t≥0,0≤μ<μ¯,that satisfy
(49)0≤ξμ<t∗<ωμ<N−pα+1p−1.(see [1]).On the other hand, the functionft has the unique minimal point
(50)t∗=N−pα+1pand is increasing on t∗,+∞. Thus, if γ≥N−pωμ, i.e., ωμ≥N−γ/p we have
(51)0=fωμ≥fN−γp.Therefore, the equalities (48) and (51) imply that
(52)0≤μ<μ~≔N−p2α+1−γ1−ppN−γpp−1.Consequently,(53)N−p2α+1−γ1−p≥0,p2α+1−Np−1≤γ<pandα<N−pp2Next, we show thatμ~≤μ¯. For any γ≥0, let
(54)μ~≔Ψγ=N−p2α+1−γ1−ppN−γpp−1.We can show easily that sinceΨ is increasing on 0,pα+1 and decreasing on pα+1,+∞ and Ψpα+1=μ¯. So, μ~≤μ¯.Lemma 8.
Let1<p<N, 0≤α<N−p/p2,α≤β<α+1,0≤γ<p,b>0,0<λ<bλ1, and 0≤μ<μ∗. Assume that q=p∗−p and 0<a<S−q+p/p, or q<p∗−p and a>0. Then, supt≥0Iλtzε<C∗.Proof.
We define the following functions(55)gt=Iλtzε=aq+ptq+pzεq+p+bptpzεp−1p∗tp∗∫Ωzεp∗xp∗βdx−λptp∫Ωzεpxγdx=aq+ptq+pzεq+p+bptpzεp−1p∗S−p∗pzep∗tp∗−1p∗∫Ωzεp∗xp∗βdx−S−P∗pzεp∗tp∗−λptp∫Ωzεp∗xγdx,ht=−1p∗S−p∗pzep∗tp∗+aq+ptq+pzεq+p+bptpzεp.
Note thatlimt⟶∞gt=−∞ and gt>0 when t is close to 0, so that supt≥0gt is attained for some Tε>0. Furthermore, from g′Tε=0 it follows that
(56)−Tεp∗−p∫Ωzεp∗xp∗βdx+aTεqzεq+p+bzεp−λ∫Ωzεpxγdx=0.Therefore,
(57)Tεp∗−pΩzεp∗xp∗βdx=aTεqzεq+p+bzεp−λ∫Ωzεpxγdx≥b−λλ1zεp.
Chooseε small enough so that by (47) we have Tε≥t0 for some t0>0.Besides, it holds(58)Tεp∗−q+p∫Ωzεp∗xp∗βdx=azεq+p+bTεqzεp−λTεq∫Ωzεpxγdx≤azεq+p+bTεqzεp.Forq<p∗−p,a>0,b>0 we have by (47)
(59)Tεp∗−q+pΩzεp∗xp∗βdx≤azεq+p+bzεpt0q.Then, forε small enough, the above estimates yield Tε<t0′ for some t0′>0 (independently of ε).Forq=p∗−p, 0<a<S−q+p/p, b>0 and for ε small enough we have by (56),
(60)Tεq=bzεp−λ∫Ωzεp/xγdx∫Ωzεp∗/xp∗βdx−azεq+p,which implies that Tε is bounded above for all ε>0, that is, there exists a positive real number t1′>0 (independently of ε).Now, we estimategTε. It follows from h′t=0(61)−S−p∗pzεp∗−ptp∗−p−atqzεq−b=0.Sety=tqS−q/pzεq and σ=p∗−p/q≥1. Then, by (61) the definition of f we get
(62)−S−1yσ−aSqpy−b=−fy=0,which implies from (26) and the proof of Lemma 6 that fy∗=0. Therefore, h′t∗=0, where t∗=S1/pzε−1y∗1/q. As fy is concave then h′t is convex and so,
(63)maxt≥0ht=ht∗=−1p∗S−p∗pzεp∗t∗p∗+aq+pzεq+pt∗q+p+bpzεpt∗p.Byh′t∗=0, we have
(64)S−p∗pzεp∗t∗p∗=azεq+pt∗q+p+bzεpt∗p.So, from (64) we deduce that
(65)maxt≥0ht=−1p∗azεq+pt∗q+p+bzεp∗t∗p∗+aq+pzεq+pt∗q+p+bpzεpt∗p=a1q+p−1p∗t∗q+pzεq+p+b1p−1p∗t∗pzεp=a1q+p−1p∗Sq+ppy∗q+pp+b1p−1p∗Sy∗pq=C∗.Consequently, by (47)
(66)supt≥0Iλtzε≤supt≥0ht+1p∗S−p∗pzεp∗−∫Ωzεp∗xp∗βdxt1p∗+−λpt0pΩzεpxγdx≤C∗+Oεpωμ+pα+1−N+−Cεpα+1−γifγ>N−pωμCεpωμ−N+pα+1lnεifγ=N−pωμ<C∗,which is the desired result.Now, we can proof the existence of a positive solution.Proof of Theorem 1.
We verify that the functionalIλ satisfies the mountain pass geometry.
Note thatIλ0=0. Let u∈DαΩ\0, by Sobolev and Young inequalities, it holds that
(67)Iλu=aq+puq+p+bpup−1p∗∫Ωup∗xp∗βdx−λp∫Ωupxγdx≥−1p∗S−p∗/p∥u∥p∗+aq+puq+p+1pb−λλ1up.
Letρ=u, since a>0,b>0 and from (67), one has
(68)Iλu≥−1p∗S−p∗/pρp∗+aq+pρq+p+b−λλ1ρp.Then, we need to consider the following cases.Caseiq<p∗−p and a>0,b>0,0<λ<bλ1.Asq+p<p∗ there exists a sufficiently small positive numbers ρ1 and δ1 such that
(69)Iλu≥δ1>0,withu=ρ1.SinceIλtu⟶−∞ as t⟶+∞ there exists t1>0 such that t1u>ρ1 and Iλt1u<0.Caseiiq=p∗−p and 0<a<S−q+p/p,b>0,0<λ<bλ1.In this case, we haveq+p=p∗, and then, from (68), one has
(70)Iλu≥a−S−q+p/pq+pρq+p+bλ1−λpλ1ρp.As0<a<S−q+p/p there exist ρ2,δ2>0 such that
(71)Iλu≥δ2>0,withu=ρ2.On the other hand, using (47) and taking ε1>0 small enough, we get
(72)Iλtzε≤atq+pq+pzεq+p+btppzεp−tp∗p∗∫Ωzεp∗xp∗βdx≤1q+paSq+pp−1Sq+pptq+p+bpSq+pptp+Oεpωμ+pα+1−Nfor all ε∈0,ε1. Then, as 0<a<S−q+p/p, it follows from the above inequality, Iλtzε⟶−∞ as t⟶+∞. Thus, choosing t2>0 sufficiently large such that t2zε>ρ2 and Iλt2zε<0.Set(73)c=infγ∈Γmaxt∈0,1Iλγt,where
(74)Γ=γ∈C0,1,DαΩ,γ0=0,γ1=t∗zε,t∗=t1ifq<p∗−panda>0t2ifq=p∗−pand0<a<S−q+pp.By the Mountain Pass Theorem, there exists a Palais-Smale sequenceun at level c. Using Lemma 5, we have that un has a subsequence, still denoted by un, such that un⇀u in DαΩ as n⟶+∞. Hence, from Lemmas 7 and 8, we have un⟶u in DαΩ as n⟶+∞,. Hence, Iλ′u=0 and Iλu=c>0. So, as c>0=Iλ0, we can conclude that u is a nonzero solution of (1) with positive energy. Now, we show that u>0. Because
(75)0=Iλ′u,u−=auq+b∫Ω∇up−2xpα∇u∇u−−μup−2xpα+1uu−dx−∫Ωup∗−2xp∗βuu−dx+−λ∫Ωup−2xγuu−dx≥auq+b∫Ω∇u−pxpα−μu−pxpα+1dx+∫Ωup∗xp∗βdx+λ∫Ωupxγdx≥bu−p,which implies that u−=0. By the strong maximum principle one has u>0. This completes the proof of Theorem 1.


## 4. Conclusion 1

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem. Our results and setting were more general and delicate, it difficult to obtain the solution in the degenerate case. Our technique was based on variational methods and concentration compactness argument and we needed to estimate the energy levels. We have shown the existence result for our problem (1) if 1<p<N, 0≤α<N−p/p2,α≤β<α+1,p2α+1−N/p−1≤γ<p,b>0,0<λ<bλ1, and 0≤μ<μ∗ with
(76)μ∗=N−p2α+1+γ1−ppN−γpp−1,and the problem (1) has a positive solution in the following cases: (1) q=p∗−p and 0<a<S−p+q/p, 2) q<p∗−p and a>0.



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*Source: 7474777-2022-03-09.xml*