# Onp-Laplace Equations with Singular Nonlinearities and Critical Sobolev Exponent

Authors: Mohammed El Mokhtar ould El Mokhtar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we deal withp-Laplace equations with singular nonlinearities and critical Sobolev exponent. By using the Nehari manifold, Mountain Pass theorem, and Maximum principle theorem, we prove the existence of at least four distinct nontrivial solutions.


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## Body

## 1. Introduction

LetΩ a bounded smooth domain in ℝNN≥3, and consider the following p-Laplace equations with singular nonlinearities
(1)Δpu=uq−1u+λu−1−βxαuinΩu=0on∂Ω,where() 1<p<N, 0∈Ω,λ>0, and q+1=p∗ with p∗≔pN/N−p is the critical Sobolev exponent, 0≤α<Nq+β/q, 0<β<1 and Δpu≔div∇up−2∇u is the p-Laplace operator which is degenerate if p>2 and singular if p<2.In recent years, researchers have been interested in studying problems of the type:(2)Δpu=uq−1u+λfx,u,inΩu=0on∂Ω,where Ω is a bounded smooth domain in ℝNN≥3, 0∈Ω,λ>0, and q+1=p∗ with p∗≔pN/N−p is the critical Sobolev exponent and f is a suitable function containing singularities on x (see [1–4] and references therein). For p=2 and after the work of Brézis and Nirenberg [5], Problem (2) has studied by many authors (see, e.g.,[6–18]). Problem (2) becomes the well-known Brézis and Nirenberg problem and is studied extensively in [19]. Ding and Tang in [20] studied the existence of positive solutions with N≥3 and fx,u satisfying (AR) condition in the case λ=1. Very recently, M. E. O. El Mokhtar et al. [21] considered Problem (1) with p=2.The term represented by the functionfx,u≔x−αuu−β with 0<β<1 is the key to this famous work because we will allow us to combine the perturbation with the variational methods to overcome shortcomings in the form of singularity. He is well known in the scientific literature that the problems dealt with in applied mathematics have their origins in different fields we will cite as example: heterogeneous chemical catalysis, kinetic chemical catalysis, heat induction or electrical induction, non-Newtonian fluid theory, and viscous fluid theory (see, e.g., [22–26]).We encounter Problem (1) in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mapping, in the generalized reaction-diffusion theory, in the turbulent flow of a gas in a porous medium, and in the non-Newtonian fluid theory (see [27–30]).Before giving our main results, we state here some definitions, notations, and known results.We denote byH=W01,pΩ\0 with respect to the norms
(3)u=∫Ω∇updx1/p.We consider the following approximation equation:(4)Δpu=uq−1u+λxαu+θβinΩu=0on∂Ω,for any θ>0. The energy functional of (4) Eθ is defined by
(5)Eθu≔1pup−1q+1∫Ωu+q+1dx−λ1−β∫Ωu++θ1−β−θ1−βxαdx.for allu∈H, where u±=maxu±,0.We know thatEθ is a C1−function on H=H01Ω.A pointu∈H is a weak solution of Equation (1) if it satisfies
(6)Eθ′u,φ≔∫Ω∇up−2∇u∇φdx−∫Ωu+q−1uφdx−λ∫Ωφu++θβxαdx=0,forallφ∈H.Here .,. denotes the product in the dualityH′, HH′dualofH.Let(7)S≔infu∈H\0up∫Ωuq+1dxp/q+1.From [3], S is achieved.Letλ∗ and λ∗∗ be positive numbers such that
(8)λ∗≔q+1−pq+βΨp−1+βq+βSq+1/pp−1+β/q+1−pand(9)λ∗∗≔q−1q+βΨ1−βp1/pp−1+βq+βp−1/q−1Spq+1−q−11+β/q−1where,(10)Ψ=2πN/2q+βNΓN/2q+β−αq+1q+β/q+1R0N/q+1q+β−α>0,with(11)0≤α<Nq+1q+β.The main results are concluded as the following theorems.Theorem 1.
Assume thatN≥3, 0≤α<N/q+1q+β, 0<β<1, and λ verifying 0<λ<λ∗; then, the system (1) has at least one positive solution.Theorem 2.
In addition to the assumptions of the Theorem1, there exists Λ∗∈0,λ∗∗ such that if λ satisfying 0<λ<Λ∗, then (1) has at least two positive solutions.Theorem 3.
Under the assumptions of Theorem2 then, there exists a positive real λ∗∗ such that if λ satisfies 0<λ<λ∗∗, then (1) has at least four nontrivial solutions.This paper is organized as follows. In Section2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last section, we prove Theorem 3.


## 2. Preliminaries

Definition 4 (see [31]).
Letc∈ℝM be a Banach space and E∈C1M,ℝ.
(i)
unn is a Palais-Smale sequence at level c (in short PSc) in M for E(12)Eun=c+on1andE′un=on1,where on1 tends to 0 as n goes at infinity
(ii)
We say thatE satisfies the PSc condition if any PSc sequence in M for E has a convergent subsequence


### 2.1. Nehari Manifold [32, 33]

It is well known thatE is of class C1 in H and the solutions of (1) are the critical points of E which is not bounded below on H. Consider the following Nehari manifold:
(13)W=u∈H\0:E′u,u=0,Note thatW contains every nontrivial solution of the problem (1). Thus, u∈W if and only if u∈H\0 and
(14)up−∫Ωu+q+1dx−λ∫Ωu++θ1−β−θ1−βxαdx=0.In our work, we research the critical points as the minimizers of the energy functional associated to the problem (1) on the constraint defined by the Nehari manifold.In order to obtain the first positive solution, we give the following important lemmas.Lemma 5.
Eθ is coercive and bounded from below on W.Proof.
LetR0>0 such that Ω⊂B0,R0=x∈ℝN:x<R0. If u∈W, then by (14) and the Hölder inequality, we obtain
(15)∫Ωu+1−βxαdx≤C1u1−β(16)withC1=ΨS1−1−β/p,
where,(17)S1≔infu∈H\0up∫Ωu1−β/xαdxp/1−β.
Therefore, we obtain that(18)Eθu=q+1−p/p1+qup+−λq+β/q+11−β∫Ωu+1−βxαdx≥q+1−p/pq+1up+−λq+β/q+11−βΨu1−βS1−1−β/p,for 0≤α<N/q+1q+β.
Thus,Eθ is coercive and bounded from below on W.Define(19)ϕu=Eθ′u,u.Then, foru∈W(20)ϕ′u,u=pup−q+1∫Ωq+1u+q+1dx+−λ1−β∫Ωu++θ1−β−θ1−βxαdx=p−1+βup−q+β∫Ωu+q+1dx=λq+β∫Ωu++θ1−β−θ1−βxαdx−q+1−pup.SplittingW in three parts, we set
(21)W+=u∈W:ϕ′u,u>0W0=u∈W:ϕ′u,u=0W−=u∈W:ϕ′u,u<0.We have the following results.Lemma 6.
Suppose thatu0 is a local minimizer for EθonW.Ifu0∉W0, then u0 is a critical point of Eθ.Proof.
Ifu0 is a local minimizer for Eθ on W, then u0 is a solution of the optimization problem:
(22)minu∈H\0/ϕu=0Eθu.
Hence, there exists a Lagrange multipliersμ∈ℝ such that
(23)Eθ′u0=μϕ′u0inH′
Thus,(24)Eθ′u0,u0=μϕ′u0,u0.
Butϕ′u0,u0≠0, since u0∉W0. Hence, μ=0. This completes the proof.Lemma 7.
There exists a positive numberλ∗ such that for all λ verifying
(25)0<λ<λ∗,we have W0=∅.Proof.
Let us reason by contradiction.
Suppose thatW0≠∅ for all λ such that 0<λ<λ∗. Then, by (20) and for u∈W0, we have
(26)p−1+βup−q+β∫Ωu+q+1dx=0,λq+β∫Ωu++θ1−β−θ1−βxαdx−q+1−pup=0.
Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain(27)u≥Sq+1/pq+1−pq+βp−1+β1/q+1−p,(28)u≤λq+βq+1−pΨ1/p−1+βS1β−1/pp−1+β.
From (26) and (27), we obtain λ≥λ∗, which contradicts our hypothesis.AsW0=∅ then W=W+∪W−. Define
(29)c≔infu∈WEθu,c+≔infu∈W+Eθuandc‐≔infu∈W‐Eθu.For the sequel, we need the following Lemma.Lemma 8.
(i) For allλ such that 0<λ<λ∗, one has c≤c+<0
(ii) For allλ such that 0<λ<λ∗∗ there exists C0>0 such hat c−>−λp/p−1+βC0Proof.
(i)
Letu∈W+. By (20), we have(30)p−1+βq+βup>∫Ωu+q+1dxand so
(31)Eθu=−p+β−1p1−βup+q+βq+11−β∫Ωu+q+1dx<p+β−1p−q−1p1−βq+1up<0,since p+1>2 and 0<β<1. Then, we conclude that c≤c+<0.
(ii)
Letu∈W−. By(20), we get(32)p−1+βq+βup<∫Ωu+q+1dx.
By Sobolev embedding theorem, we obtain(33)∫Ωu+q+1dx≤S−q+1/puq+1.
This implies(34)u>Sq+1/pq+1−pp−1+βq+β1/q+1−p,forallu∈W−.
By the proof of Lemma5, we have
(35)Eθu≥upq−1pq+1+−λu1−βq+βq+11−βΨS1−β/p≥q−ppqp−1+βq+βp/q−1Sq+1/q−1−λpq+βpq+1q−11−βΨpS1−β/p.
Thus, for allλ such that 0<λ<λ∗∗ we have Eθu≥−λp/p−1+βC0 with
(36)C0=1β−1−1q+1p−1+βpΨq+βq−p+11−β/p−1+β.
As in [34] we have the following result.Proposition 9.
(i) For allλ such that 0<λ<λ∗, there exists aPSc+ sequence in W+
(ii) For allλ such that 0<λ<λ∗∗, there exists a PSc− sequence in W− and for each u∈H\0.
Define(37)tM≔tmaxu=p−1+βupq+p∫Ωu+q+1dx1/q+1−p>0.Lemma 10.
Suppose that0<λ<λ∗. For each u∈H\0, there exists unique t+ and t− such that 0<t+<tM<t−,t+u∈W+, and t−u∈W−,(38)Eθt+u=infEtufor0≤t≤tM,Eθt−u=supEtufort≥0.Proof.
With minor modifications, we refer the reader to [34].


## 2.1. Nehari Manifold [32, 33]

It is well known thatE is of class C1 in H and the solutions of (1) are the critical points of E which is not bounded below on H. Consider the following Nehari manifold:
(13)W=u∈H\0:E′u,u=0,Note thatW contains every nontrivial solution of the problem (1). Thus, u∈W if and only if u∈H\0 and
(14)up−∫Ωu+q+1dx−λ∫Ωu++θ1−β−θ1−βxαdx=0.In our work, we research the critical points as the minimizers of the energy functional associated to the problem (1) on the constraint defined by the Nehari manifold.In order to obtain the first positive solution, we give the following important lemmas.Lemma 5.
Eθ is coercive and bounded from below on W.Proof.
LetR0>0 such that Ω⊂B0,R0=x∈ℝN:x<R0. If u∈W, then by (14) and the Hölder inequality, we obtain
(15)∫Ωu+1−βxαdx≤C1u1−β(16)withC1=ΨS1−1−β/p,
where,(17)S1≔infu∈H\0up∫Ωu1−β/xαdxp/1−β.
Therefore, we obtain that(18)Eθu=q+1−p/p1+qup+−λq+β/q+11−β∫Ωu+1−βxαdx≥q+1−p/pq+1up+−λq+β/q+11−βΨu1−βS1−1−β/p,for 0≤α<N/q+1q+β.
Thus,Eθ is coercive and bounded from below on W.Define(19)ϕu=Eθ′u,u.Then, foru∈W(20)ϕ′u,u=pup−q+1∫Ωq+1u+q+1dx+−λ1−β∫Ωu++θ1−β−θ1−βxαdx=p−1+βup−q+β∫Ωu+q+1dx=λq+β∫Ωu++θ1−β−θ1−βxαdx−q+1−pup.SplittingW in three parts, we set
(21)W+=u∈W:ϕ′u,u>0W0=u∈W:ϕ′u,u=0W−=u∈W:ϕ′u,u<0.We have the following results.Lemma 6.
Suppose thatu0 is a local minimizer for EθonW.Ifu0∉W0, then u0 is a critical point of Eθ.Proof.
Ifu0 is a local minimizer for Eθ on W, then u0 is a solution of the optimization problem:
(22)minu∈H\0/ϕu=0Eθu.
Hence, there exists a Lagrange multipliersμ∈ℝ such that
(23)Eθ′u0=μϕ′u0inH′
Thus,(24)Eθ′u0,u0=μϕ′u0,u0.
Butϕ′u0,u0≠0, since u0∉W0. Hence, μ=0. This completes the proof.Lemma 7.
There exists a positive numberλ∗ such that for all λ verifying
(25)0<λ<λ∗,we have W0=∅.Proof.
Let us reason by contradiction.
Suppose thatW0≠∅ for all λ such that 0<λ<λ∗. Then, by (20) and for u∈W0, we have
(26)p−1+βup−q+β∫Ωu+q+1dx=0,λq+β∫Ωu++θ1−β−θ1−βxαdx−q+1−pup=0.
Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain(27)u≥Sq+1/pq+1−pq+βp−1+β1/q+1−p,(28)u≤λq+βq+1−pΨ1/p−1+βS1β−1/pp−1+β.
From (26) and (27), we obtain λ≥λ∗, which contradicts our hypothesis.AsW0=∅ then W=W+∪W−. Define
(29)c≔infu∈WEθu,c+≔infu∈W+Eθuandc‐≔infu∈W‐Eθu.For the sequel, we need the following Lemma.Lemma 8.
(i) For allλ such that 0<λ<λ∗, one has c≤c+<0
(ii) For allλ such that 0<λ<λ∗∗ there exists C0>0 such hat c−>−λp/p−1+βC0Proof.
(i)
Letu∈W+. By (20), we have(30)p−1+βq+βup>∫Ωu+q+1dxand so
(31)Eθu=−p+β−1p1−βup+q+βq+11−β∫Ωu+q+1dx<p+β−1p−q−1p1−βq+1up<0,since p+1>2 and 0<β<1. Then, we conclude that c≤c+<0.
(ii)
Letu∈W−. By(20), we get(32)p−1+βq+βup<∫Ωu+q+1dx.
By Sobolev embedding theorem, we obtain(33)∫Ωu+q+1dx≤S−q+1/puq+1.
This implies(34)u>Sq+1/pq+1−pp−1+βq+β1/q+1−p,forallu∈W−.
By the proof of Lemma5, we have
(35)Eθu≥upq−1pq+1+−λu1−βq+βq+11−βΨS1−β/p≥q−ppqp−1+βq+βp/q−1Sq+1/q−1−λpq+βpq+1q−11−βΨpS1−β/p.
Thus, for allλ such that 0<λ<λ∗∗ we have Eθu≥−λp/p−1+βC0 with
(36)C0=1β−1−1q+1p−1+βpΨq+βq−p+11−β/p−1+β.
As in [34] we have the following result.Proposition 9.
(i) For allλ such that 0<λ<λ∗, there exists aPSc+ sequence in W+
(ii) For allλ such that 0<λ<λ∗∗, there exists a PSc− sequence in W− and for each u∈H\0.
Define(37)tM≔tmaxu=p−1+βupq+p∫Ωu+q+1dx1/q+1−p>0.Lemma 10.
Suppose that0<λ<λ∗. For each u∈H\0, there exists unique t+ and t− such that 0<t+<tM<t−,t+u∈W+, and t−u∈W−,(38)Eθt+u=infEtufor0≤t≤tM,Eθt−u=supEtufort≥0.Proof.
With minor modifications, we refer the reader to [34].


## 3. Proof of Theorem1

Now, taking as a starting point the work of Tarantello [35], we establish the existence of a local minimum for Eθ on W+.Proposition 11.
For allλ such that 0<λ<λ∗, the functional Eθ has a minimizer u0+∈W+, and it satisfies
(i)
Eθu0+=c+(ii)
u0+ is a nontrivial solution of (1)Proof.
If0<λ<λ∗, then by Proposition 9, (i) there exists a unPSc+ sequence in ;, thus, it bounded by Lemma 5. Then, there exists u0+∈H, and we can extract a subsequence which will denoted by un such that
(39)un⟶u0+weaklyinHun⟶u0+stronglyinL1−βΩ,x−α.un⟶u0+a.einΩ
By (15) and (39), we have
(40)limn⟶∞∫Ωun+θ1−βxαdx=∫Ωu0++θ1−βxαdx+o1.
Thus, by (39), u0+ is a weak nontrivial solution of (1). Now, we show that un converges to u0+ strongly in H. Suppose otherwise. By the lower semi-continuity of the norm, if un1/2u0+ we have u0+<liminfn⟶∞un, and we obtain
(41)c≤Eθu0+−1q+1Eθ′=q+1−ppq+1/u0+p+−λq+βq+11−β∫Ωu0++θ1−βxαdx<liminfn⟶∞Eθun=c.
We get a contradiction. Therefore,un converge to u0+ strongly in H. Moreover, we have u0+∈W+. If not, then by Lemma 10, there are two numbers t0+ and t0−, uniquely defined so that t0+u0+∈W+ and t−u0+∈W−. In particular, we have t0+<t0−=1. Since
(42)ddtEθtu0+≫t=t0+=0,d2dt2Eθtu0+≫t=t0+>0,there exists t0−<t−≤t0+ such that Eθt0−u0+<Eθt+u0+. Then, we get
(43)Eθt0−u0+<Eθt−u0+<Eθt0+u0+=Eθu0+,
which contradicts the fact thatEθu0+=c+. Since Eθu0+=Eθu0+ and u0+∈W+, then by Lemma 6, we may assume that u0+ is a nontrivial nonnegative solution of (1). By the Harnack inequality, we conclude that u0+>0 (see, e.g., [29]).


## 4. Proof of Theorem2

Next, we establish the existence of a local minimum forEθ on W−. For this, we require the following Lemmas.Lemma 12.
Letun be PSc sequence for Eθ for some c∈ℝ with un⟶u in H.
Then,Eθ′u=0 and Eθu≥−λp/1+βCq,β,Ψ,S, with Cq,β,Ψ,S>0.Proof.
Letun⊂Su be a minimizing sequence for Eθ with Su is the unit sphere. By Ekeland’ s variational principle [12], we may assume Eθ′un⟶0. So un is a PSc sequence and therefore un⟶u after passing to a subsequence. Hence Eθu=c and Eθ′u=0, which implies that Eθ′u,u=0, and
(44)∫Ωu+q+1dx=up−λ∫Ωu++θ1−βxαdx.
Therefore,(45)Eθu=q−1pp+1up+−λq+βq+1p+1−β∫Ωu++θ1−βxαdx.
From (15) and considering u small enough, we get
(46)∫Ωu++θ1−βxαdx≤2πN/2q+βNΓN/2q+β−αq+1q+β/q+1R0N/q+1q+β−αu1−βS−1−β/p,which implies that
(47)Eu≥q−1pq+1up−λq+βq+1p+1−βS−1−β/pΨu1−β,with
(48)Ψ=2πN/2q+βNΓN/2q+β−αq+1q+β/q+1R0N/q+1q−1+β−α.
Setft=Dtp−λEt1−β for all t>0, with
(49)D=q−1pq+1,E=q+βq+11−βS−1−β/pΨ.
Using (46), we obtain that
(50)ft≥−λp/p−1+βCp,q,β,Ψ,Sforallt>0smallenough,where
(51)Cp,q,β,Ψ,S=D1−βEpDp/1+ββ1−β,
We haveCp,q,β,Ψ,S>0 since 0<β<1. Then, we conclude that
(52)Eu≥−λp/p−1+βCp,q,β,Ψ,S.Lemma 13.
Letλ∈0,λ∗∗; then, the functional Eθ satisfies the PSc condition in H with c<c∗, where
(53)c∗=q+1−ppq+1Sq+1/q+1−p−λp/p−1+βCp,q,β,Ψ,S.Proof.
If0<λ<λ∗∗, then by Proposition 9, (ii) there exists a un, PSc sequence in W; thus, it bounded by Lemma 5. Then, there exists u∈H, and we can extract a subsequence which will denoted by un such that
(54)un⇀uweaklyinHun⇀uweaklyinLq+1Ωun⟶ua.einΩ.
Then,u is a weak solution of (1). Let vn=un−u; then, by Brézis-Lieb [36], we obtain
(55)vnp=unp−up+on1,(56)∫Ωvnq+1dx=∫Ωunq+1−dx∫Ωuq+1dx+on1.
Since(57)Eθun=c+on1,Eθ′un=on1,and by (55) and (56), we deduce that
(58)1pvnp−1q+1∫Ωvnq+1dx=c−Eθu+on1,vnp−∫Ωvnq+1dx=on1.
Hence, we may assume that(59)vnp⟶l,∫Ωvnq+1dx⟶l.
Moreover, by Sobolev inequality, we have(60)vnp≥S∫Ωvnq+1dx.
Combining (60) and (59), we obtain
(61)l≥lp/q+1S.
Either,(62)l=0orl≥Sq+1/q−1.
Then from (58), (59), Lemma 13 and Lemma 12, we obtain
(63)c≥q−1pq+1l+Eθun≥c∗.which is a contradiction. Therefore, l=0, and we conclude that un converges to u strongly in H.
Thus,(64)EθunconvergestoEθu=casntendsto+∞.Lemma 14.
There existsv∈H and Λ∗>0 such that for all λ∈0,Λ∗, one has
(65)supt≥0Eθtv<c∗.
In particular,c<c∗ for all λ∈0,Λ∗.Proof.
Letφεx satisfies (4). Then, we have
(66)λ1−β∫Ωφε+θ1−βxαdx>0.
We consider the two functions:(67)ft≔Eθtφεandgt=tppφεp−tq+1q+1∫Ωφεq+1dx.
Then, for all for allλ∈0,λ∗∗,
(68)f0=0<c∗.
By the continuity off, there exists t0>0 such that
(69)ft<c∗,∀t∈0,t0.
On the other hand we have(70)maxt≥0gt=q−1q+βp1+βq+1Sq+1/q−1.
Then, we obtain(71)supt≥0Eθtφε<q−1q+βp1+βq+1Sq+1/q−1−λp/1+βCp,q,β,Ψ,S.
Now, takingλ>0 such that
(72)−λt01−β1−β∫Ωφε1−βxαdx<−λp/1+βCp,q,β,Ψ,S,we obtain
(73)0<λ<t01+β1−βCp,q,β,Ψ,S1+β/1−β∫Ωφε+θ1−βxαdx1+β/1−β=Λ1.
Set(74)Λ∗=minλ∗∗,Λ1.
We deduce thatc−<c∗ for all λ∈0,Λ∗; then, there exists tn>0 such that tnwn∈W− with wn satisfying (4) and for all λ∈0,Λ∗,(75)c−≤Eθtnwn≤supt≥0Etwn<c∗.Lemma 15.
For allλ such that 0<λ<Λ∗=minλ∗∗,Λ1, the functional Eθ has a minimizer u0−inW−, and it satisfies
(76)Eθu0−=c−>0,
u0− is a nontrivial solution of (1) in H.Proof.
By Proposition9 (ii), there exists a PSc− sequence un for E, in W− for all λ∈0,λ∗∗. From Lemmas 13, 18, and 8(ii), for λ∈0,Λ1, Eθ satisfies PSc− condition and c−>0. Then, we get that un is bounded in H. Therefore, there exist a subsequence of un still denoted by un and u0−∈W− such that un converges to u0− strongly in H and Eθu0−=c−>0 for all λ∈0,Λ∗.
Finally, by using the same arguments as in the proof of Proposition11 for all λ∈0,λ∗, we have that u0− is a solution of (1).Now, we complete the proof of Theorem2. By Proposition 11 and Lemma 15, we obtain that (1) has two positive solutions u0+∈W+ and u0−∈W−. Since W+∩W−=∅, then, u0+ and u0− are distinct.


## 5. Proof of Theorem3

Now, we consider the following Nehari submanifold ofW:
(77)Wρ=u∈H/0:Eθ′u,u=0andu≥ρ>0.Thus,u∈Wρ if and only if
(78)up−∫Ωu+q+1dx−λ∫Ωu++θ1−βxαdx=0,u≥ρ>0.Firstly, we need the following Lemmas.Lemma 16.
Under the hypothesis of theorem3, there exist Λ2>0 such that Wρ is nonempty for any λ∈0,Λ2.Proof.
Fixu0∈H\0, and let
(79)gt=Eθ′tu0,tu0=tpu0p−tq+1∫Ωu0q+1dx−t1−βλ∫Ωu0+θ1−βxαdx.
Clearlyg0=0 and gt⟶−∞ as t⟶+∞. Moreover, we have
(80)g1=u0p−∫Ωu0q+1dx−λ∫Ωu0+θ1−βxαdx≥u02−βu0β−Sμ−q+1/pu0q+β−λΨSβ−1/p.for t≥0, put φt=tβ−S−q+1/ptq+β; then, we obtain maxt≥0φt=φt1>0 since q>2 with t1=β/q−+βq+1/q−1Sq+1/pq−1. Thus, we obtain
(81)g1≥u01−βφt1−λΨSβ−1/p>0,if λ<S1−β/p/Ψφt1≔Λ2.
Then, there existst0>0 such that gt0=0. Thus, t0u0∈Wρ and Wρ is nonempty for any λ∈0,Λ2.Lemma 17.
There existδ,λ∗ positive real numbers such that ϕ′u,u?<−?<0, for u∈Wρ and any λ verifying
(82)0<λ<λ∗.Proof.
Letu∈Wρ, then by (14), (20) and the Holder inequality, it allows us to write
(83)ϕ′u,u=λq+βΨSβ−1/pu1−β−q−1up≤u1−βλq+βΨSβ−1/p−q−1uβ≤u1−βλq+βΨSβ−1/p−q−1ρβ.
Thus, if(84)0<λ<Λ3=q−1ρβq+βΨSβ−1/p,and choosing λ∗≔minΛ2,Λ3 with Λ2 defined in Lemma 16, then we obtain that
(85)ϕ′u,u<0,foranyu∈Wρ.Lemma 18.
Supposeq>2,β∈0,1 and 0<λ<minΛ2,Λ3,Λ4 when
(86)Λ4=q−11−βpqq+βΨ1−βpβ+1/βS1−q/p.
Then, there existε and η positive constants such that
(i)
We have(87)Eθu≥η>0foru=ε.(ii)
There existsv∈Wρ when v>ε, with ε=u, such that Eθv≤0Proof.
We can suppose that the minima ofEθ are realized by u0+ and u0−. The geometric conditions of the mountain pass theorem are satisfied. Indeed, we have the following:
(i)
By (20) and (85), we get(88)Eθu=q−1/pq+1up−λq+β/q+11−β∫Ωu++θ1−βxαdx≥q−1/pq+1up−λq+β/q+11−βΨSβ−1/pu1−β,
By exploiting the functionϕt=atp−bt1−β which achieve its maximum at the point t1=1−β/pp/β−1+pa/bp−1/q−1 such that maxt≥0ϕt=ϕt1>0 if
(89)λ<Λ4=q−11−βpq+βΨ1−βpβ/p−1+βS1−q/p,and the fact that, q>2,β∈0,1 then, we obtain that
(90)Eθu≥η>0whenε=usmall.(ii)
Lett>0, then we have for all φ∈Wρ(91)Eθtφ≔tppφp−tq+1q+1∫Ωφq+1dx−λt1−β1−β∫Ωx−αφ1−βxαdx.
Lettingv=tφ for t large enough, we obtain Eθv≤0. For t large enough, we can ensure v>ε.LetΓ and c defined by
(92)Γ≔γ:0,1⟶Wρ:γ0=u0−andγ1=u0+,c≔infγ∈Πmaxt∈0,1Eθγt.Proof of Theorem 19.
If(93)0<λ<λ∗∗≔minλ∗,Λ4,then, by the Lemma 5 and Proposition 9 (ii), Eθ verifying the Palais-Smale condition in Wρ. Moreover, from the Lemmas 6, 17, and 18, there exists uc such that
(94)Eθuc=canduc∈Wρ.
Thus,uc is the third solution of our system such that uc≠u0+ and uc≠u0−. Since (1) is odd with respect u, we obtain that −uc is also a solution of (1).
Finally, for everyθ∈0,1, problem (4) has solution uθ∈H such that Eθuθ=0. Thus, there exist θn⊂0,1 with θn⟶0 as n⟶∞. Then, we get u=limn⟶∞uθn.


## 6. Conclusion

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem on the constraint defined by the Nehari manifoldW, which are solutions of our problem. Under some sufficient conditions on coefficients of equation of (1) such that N≥3, 0≤α<Nq+β/q+1 and β∈0,1, we split W in two disjoint subsets W+ and W−; thus, we consider the minimization problems on W+ and W−, respectively. In Sections 3 and 4 we have proved the existence of at least two nontrivial solutions on Wρ for all 0<λ<λ∗∗≔minλ∗,Λ4 if N≥3 and β∈0,1.



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