# Variational Approach for the Variable-Order Fractional Magnetic Schrödinger Equation with Variable Growth and Steep Potential inℝN∗

Authors: Jianwen Zhou; Bianxiang Zhou; Liping Tian; Yanning Wang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we show the existence of solutions for an indefinite fractional Schrödinger equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents and steep potential. By using the decomposition of the Nehari manifold and variational method, we obtain the existence results of nontrivial solutions to the equation under suitable conditions.


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## Body

## 1. Introduction

In this paper, we investigate the existence of solutions of the following concave-convex fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator involving variable exponents:(1)−ΔAs·u+Vλxu=fxu∣qx−2u+gxupx−2uinℝN,where N≥1,s·:ℝN×ℝN→0,1, is a continuous function, −ΔAs· is the variable-order fractional magnetic Laplace operator, the potential Vλx=λV+x−V−x with V±=max±V,0, λ>0 is a parameter, and the magnetic field is A∈C0,αℝN,ℝN with α∈0,1,p,q∈CℝN and u:ℝN→ℂ. In [1], the fractional magnetic Laplacian has been defined as
(2)−ΔAsux=limr→0∫Brcxux−eix−y·Ax+y/2uyx−yN+2sdy,for x∈ℝN. In [2], the variable-order fractional Laplace −Δs· is defined as for each x∈ℝN,
(3)−Δs·ux=2P.V∫ℝNux−uyx−yN+2sx,ydy,along any u∈C0∞Ω. Inspired by them, we define the variable-order fractional magnetic Laplacian −ΔAs· as for each x∈ℝN,
(4)−ΔAs·ux=limr→0∫Brcxux−eix−y·Ax+y/2uyx−yN+2sx,ydy.Sinces· is a function, magnetic field A∈C0,αℝN,ℝN with α∈0,1, we see that operator −ΔAs· is a variable-order fractional magnetic Laplace operator. Especially, when s·≡constant,−ΔAs· reduces to the usual fractional magnetic Laplace operator. When s·≡constant,A=0,−ΔAs· reduces to the usual fractional Laplace operator. Very recently, when A=0,V−x=0 and fx,gx≡constant, authors in [2] are given some sufficient conditions to ensure the existence of two different weak solutions, and used the variational method and the mountain pass theorem to obtain the two weak solutions of problem (5) which converge to two solutions of its limit problems, and the existence of infinitely many solutions to its limit problem:
(5)−Δs·u+λVxu=αupx−2u+βuqx−2uinΩ,u=0inℝN\Ω.In addition, authors studied the multiplicity and concentration of solutions for a Hamiltonian system driven by the fractional Laplace operator with variable-order derivative in [3]. For s·=1, px, qx≡constant, and A=0, in [4], authors obtained the multiplicity and concentration of the positive solution of the following indefinite semilinear elliptic equations involving concave-convex nonlinearities by the variational method:
(6)−Δu+Vλxu=fxu∣q−2u+gxup−2uinℝN,u≥0inℝN.Fors·, px, qx≡constant, and A=0, in [5], under appropriate assumptions, Peng et al. obtained the existence, multiplicity, and concentration of nontrivial solutions for the following indefinite fractional elliptic equation by using the Nehari manifold decomposition:
(7)−Δαu+Vλxu=axuq−2u+bxup−2uinℝN,u≥0inℝN.In [1], by using the Nehari manifold decomposition, authors studied the concave-convex elliptic equation involving the fractional order nonlinear Schrödinger equation:
(8)−ΔAsu+Vλxu=fxuq−2u+gxup−2uinℝN.Some sufficient conditions for the existence of nontrivial solutions of equation (8) are obtained. Nevertheless, only a few papers see [6–12] deal with the existence and multiplicity of fractional magnetic problems. Some papers see [8, 13–16] deal with the solvability of Kirchhoff problems. Inspired by above, we are interested in the existence and multiplicity of solutions to problem (1) with variable growth and steep potential in ℝN. As far as we know, this is the first time to study the multiplicity of nontrivial solutions of the indefinite fractional elliptic equation driven by the variable-order fractional magnetic Laplace operator with variable exponents and steep potential in ℝN. This result was improved in the recent paper [1].It is worth mentioning that in this paper, we not only obtain the existence and multiplicity results of nontrivial solutions of the variable-order fractional magnetic Schrodinger equation with variable growth and steep well potential inℝN but also our main results are based on the study for the decomposition of Nehari manifolds. On the one hand, relative to [1], we extend the exponent to variable exponent, thus introducing the variable exponent Lebesgue space. In addition, compared with [2], we extend the range of px to 2,∞ and the research range from the bounded region Ω to the whole space ℝN. On the other hand, if we want to find the nontrivial solution of the equation (1) by the variational method, we need some geometry, such as a mountain structure and a link structure. However, the energy functional of equation (1) does not have the mountain structure. In order to overcome this obstacle, we seek another method, the Nehari manifold. By decomposing the Nehari manifold into three parts, we obtain the existence of nontrivial solutions of each part.Inspired by the above works, we assume(S1) There exist two constants 0<s0<s1<1 such that s0<sx,y<s1 for all x,y∈ℝN×ℝN.(S2) s· is symmetric, that is, sx,y=sy,x for all x,y∈ℝN×ℝN.(V1) V+ is a continuous function on ℝN and V−∈LN/2ℝN.(V2) There exists k>0 such that the set V+<k=x∈ℝN:V+x<k is a nonempty and has finite measure. In addition, M2∣V+<k∣<1, where ∣·∣ is the Lebesgue measure and M is the best Sobolev constant (see Lemma 9).(V3) Ω=x∈ℝN,V+x=0 is nonempty and has a smooth boundary with Ω¯=intx∈ℝN,V+x=0.(V4) There exists a constant ϑ0>1 such that
(9)infu∈DAs·ℝN,ℂ0∫ℝ2Nux−eix−y·Ax+y/2uy2/x−yN+2sx,ydxdy+λ∫ℝNV+u2dx∫ℝNV−u2dx≥ϑ0,for all λ>0, where DAs·,ℝN,ℂ is the Hilbert space related to the magnetic field A (see Section 2).(V5) ∣V+<k∣>maxA1,A2,A3,A4>0 where
(10)A1=p−−2q−2p−−q−∥f∥L∧2/2−qxℝN2q+−22−q+p+−q+∥g∥∞22−p+ϑ0−1θϑ02p+−q+p+−2q+−2,A2=p−−2q−2p−−q−∥f∥L∧2/2−qxℝN2p−−22−p+2−q+ϑ0−1ϑ02p−−q+2−p+2−q+2−q+p+−q+∥g∥∞22−p+·θ2p+−q+p++q+p−−2q+2−p+2−q+M2p+−p−2−p+,A3=p−−2q−2p−−q−∥f∥L∧2/2−qxℝN2q−−2ϑ0−1θϑ02p+−q−p+−2q−−22−q+p+−q+∥g∥∞22−p+,A4=p−−2q−2p−−q−∥f∥L∧2/2−qxℝN2p−−22−p+2−q−2−q+p+−q+∥g∥∞22−p+ϑ0−1ϑ02p−−q−2−p+2−q−·θ2p+−q−p++q−p−−2q−2−p+2−q−M2p+−p−2−p+.To the best of our knowledge, this type of hypothesis is the first introduced by Bartsch and Wang in [17]. In addition, we recall the potential Vλ satisfied the conditions V1−V3 as the steep well potential.Concerningpx,qx and fx,gx, we suppose(H1) A measurable function p:ℝN→2,+∞ satisfy
(11)2<p−≔essinfx∈ℝNpx≤p+=esssupx∈ℝNpx<∞.(H2) A measurable function q:ℝN→1,2 satisfy
(12)1<q−≔essinfx∈ℝNqx≤q+=esssupx∈ℝNqx<2.(H3) f∈L2/2−qxℝN,ℂ and ∥f∥L2/2−qxℝN,ℂ>0, where LpxℝN,ℂ will be given in Section 2.(H4) g∈L∞ℝN,ℂ and ∥g∥∞≔∥g∥L∞ℝN,ℂ>0.(13)In what follows, it will always be assumed that the hypothesisS2 holds. Then, we will give the following definition of weak solutions for problem (1).Definition 1.
We say thatu∈Xλ is a weak solution of equation (1), if
(13)R∫ℝ2Nux−eix−y·Ax+y/2uyvx−eix−y·Ax+y/2vy¯x−yN+2sx,ydxdy+λR∫ℝNV+uv¯dx−R∫ℝNV−uv¯dx−R∫ℝNfxuqx−2u+gxupx−2uv¯dx=0,for any v∈Xλ, where Xλ will be given in Section 2.
Our main results are as follows.Theorem 2.
Under (V1)–(V4), (H1)–(H4), and (H5), there exists a nonempty open set Ωg⊂Ω such that gx>0 in Ωg. Then, equation (1) allows at least a nontrivial solution for all λ>1/kM∣V+<k∣.Theorem 3.
Suppose that (S1), (S2), (V1)–(V2), and (H1)–(H4) are satisfied. Then, there exists λ∗≥0 such that for every λ>λ∗, equation (1) has at least two nontrivial solutions.Remark 4.
Generally speaking, ifs·:ℝN×ℝN→0,1 is a continuous function, magnetic field A∈C0,αℝN,ℝN with α∈0,1, then the variable-order fractional magnetic Laplacian can be defined as for each given u∈C0∞ℝN,ℂ,
(14)−ΔAs·u,v=R∫ℝ2Nux−eix−y·Ax+y/2uyvx−eix−y·Ax+y/2vy¯x−yN+2sx,ydxdy,along any v∈C0∞ℝN,ℂ.


## 2. Preliminaries and Notations

For the reader’s convenience, we first review some necessary definitions that we are later going to use of variable exponent Lebesgue spaces. We refer the reader to [2, 3, 18–20] for details. Furthermore, we give the variational setting for equation (1) and some preliminary results.Denote(15)p+=esssupx∈ℝNpx,p−=essinfx∈ℝNpx.Ifp+<∞, then p is said to be bounded. If 1/px+1/p′x=1, then p′x=px/px−1 is called the dual variable exponent of px. The variable exponent Lebesgue space can be defined as
(16)LpxℝN,ℂ=u:ℝN→ℂisameasurablefunction;ρpxu=∫ℝNuxpxdx<∞with the norm
(17)∥u∥Lp·ℝN,ℂ=infμ>0:ρpxμ−1u≤1,then LpxℝN,ℂ is a Banach space. When p is bounded, we have
(18)min∥u∥LpxℝN,ℂp−,∥u∥LpxℝN,ℂp+≤∫ℝNuxpxdx≤max∥u∥LpxℝN,ℂp−,∥u∥LpxℝN,ℂp+.For bounded exponent, the dual spaceLpxℝN,ℂ′ can be identified with Lp′xℝN,ℂ, where p′x is called the dual variable exponent of px. Especially,
(19)L2ℝN,ℂ=u:ℝN→ℂisameasurablefunction;∫ℝNux2dx<∞with the real scalar product u,vL2ℝN,ℂ≔R∫ℝNuv¯dx, for all u,v∈L2ℝN,ℂ. By Lemma 11, 20 of [20] and ∥·∥LpxℝN,ℂ=∥∣·∣∥LpxℝN,ℝ, we know that in the variable exponent Lebesgue space, the Ho¨lder inequality is still valid. For all u∈LpxℝN,ℂ,v∈Lp′xℝN,ℂ with px∈1,∞, the following inequality holds
(20)∫ℝNuvdx≤1p−+1p′−∥u∥LpxℝN,ℂ∥v∥Lp′xℝN,ℂ≤2∥u∥LpxℝN,ℂ∥v∥Lp′xℝN,ℂ.Define(21)Ds·ℝN,ℂ=u∈L2ℝN,ℂ:∫ℝN∫ℝNux−uy2x−yN+2sx,ydxdy<∞.EquipDs·ℝN,ℂ with the inner product
(22)u,vs·=∫ℝN∫ℝNux−uyvx−vyx−yN+2sx,ydxdy+∫ℝNuxvxdx,and the corresponding norm ∥u∥s·2=u,us·. Especially, if s·≡constant, then the space Ds·ℝN,ℂ is the usual fractional Sobolev space DsℝN,ℂ.Lemma 5 (see [3] Lemma 5).
Letp∈2,2s0∗, 2s0∗=2N/N−2s0, if N>2; 2s0∗=∞ if N≤2. The embedding Ds1ℝN,ℝ↪Ds·ℝN,ℝ↪Ds0ℝN,ℝ↪LpℝN,ℝ are continuous.
For each functionu:ℝN→ℂ, set
(23)us·,A2≔∫ℝ2Nux−eix−y·Ax+y/2uy2x−yN+2sx,ydxdy,and the corresponding norm is defined as ∥u∥s·,A2=∥u∥L2ℝN,ℂ2+us·,A2. Set D be the space of measurable functions u:ℝN→ℂ such that ∥u∥s·,A<∞; then, D,·,·s·,A is a Hilbert space. If we let DAs·ℝN,ℂ as the closure of Cc∞ℝN,ℂ in D, then DAs·ℝN,ℂ is a Hilbert space.Lemma 6.
For each compact subsetW⊂ℝN, the embedding DAs·ℝN,ℂ↪Ds·W,ℂ is continuous.Proof.
Fixed any compact subsetW⊂ℝN, for any u∈Ds·W,ℂ, we have
(24)∥u∥Ds·W,ℂ2=∫Wux2dx+∫W∫Wux−uy2x−yN+2sx,ydxdy≤∫ℝNux2dx+2∫W∫Wux−eix−y·Ax+y/2uy2x−yN+2sx,ydxdy+2∫W∫Wuy2eix−y·Ax+y/2−12x−yN+2sx,ydxdy≤2∫ℝNux2dx+2∫ℝN∫ℝNux−eix−y·Ax+y/2uy2x−yN+2sx,ydxdy+2∫W∫Wuy2eix−y·Ax+y/2−12x−yN+2sx,ydxdy≤2∥u∥s·,A2+2J,where
(25)J≔∫W∫Wuy2eix−y·Ax+y/2−12x−yN+2sx,ydxdy=∫Wuy2∫W∩∣x−y∣>1eix−y·Ax+y/2−12x−yN+2sx,ydxdy+∫Wuy2∫W∩∣x−y∣≤1eix−y·Ax+y/2−12x−yN+2sx,ydxdy=J1+J2.
Since∣eit−1∣≤2, we have
(26)J1≤4∫Wuy2∫W∩∣x−y∣>11x−yN+2sx,ydxdy≤4∫Wuy2∫W∩∣x−y∣>11x−yN+2s0dxd=4∫Wuy2∫W∩∣z∣>11zN+2s0dzdy≤C1∫Wuy2dy≤C1∫ℝNuy2dy=C1∥u∥L2ℝN,ℂ2.
By Lemma6 of [21], we know that A is locally bounded, and W⊂ℝN is compact, eix−y·Ax+y/2−12≤C2x−y2,for∣x−y∣≤1,x,y∈W. Thus, we obtain
(27)J2=∫Wuy2∫W∩∣x−y∣≤1eix−y·Ax+y/2−12x−yN+2sx,ydxdy≤∫Wuy2∫W∩∣x−y∣≤1C2x−yN+2sx,y−2dxdy≤∫Wuy2∫W∩∣x−y∣≤1C2x−yN+2s1−2dxdy≤C2∫Wuy2∫W∩∣z∣≤11zN+2s1−2dzdy≤C3∫Wuy2dy≤C3∫ℝNuy2dy=C3∥u∥L2ℝN,ℂ2.
By (24)-(27), we can easily get that
(28)∥u∥Ds·W,ℂ2≤2∥u∥s·,A2+2C1∥u∥L2ℝN,ℂ2+2C3∥u∥L2ℝN,ℂ2≤C4∥u∥s·,A2,which implies that the embedding DAs·ℝN,ℂ is continuously embedded into Ds·W,ℂ.Through the above lemma, we know thatDAs·ℝN,ℂ↪Ds·W,ℂ, and from Theorem 2.1 of [2], we know that for Ω be a bounded subset of ℝN and p:Ω¯→1,∞ is continuous functions, Ds·Ω,ℂ is continuously embedded into LpxΩ,ℂ, so we seek another method to prove the size relationship between ∫ℝNuxpxdx, ∫ℝNuxqxdx, and ∥u∥λ.Lemma 7 (see [6] Lemma 10).
For everyu∈DAs·ℝN,ℂ, it holds ∣u∣∈Ds·ℝN,ℝ. More precisely, °Í(29)∥∣u∣∥s·≤∥u∥s·,A,foreveryu∈DAs·ℝN,ℂ.Remark 8 (see [6] Remark 9).
There holds(30)∣ux−eix−y·Ax+y2uy∣≥ux∣−∣uy,fora.e.x,y∈ℝN.Lemma 9.
Letp∈2,2s0∗, where 2s0∗=2N/N−2s0ifN>2; 2s0∗=∞ifN≤2. DAs·ℝN,ℂ is continuously embedded into LpℝN,ℂ. Moreover, if s0∈1/2,1, then DAs·ℝN,ℂ can be continuously embedded into L∞ℝN,ℂ; that is, there exists a constant M>0 such that
(31)∥u∥∞≤M∥u∥s·,A.Proof.
By Lemma7, we know that for every u∈DAs·ℝN,ℂ, it holds ∣u∣∈Ds·ℝN,ℝ. By Lemma 5, we know that for Ds·ℝN,ℝ↪LpℝN,ℝ is continuous. In light of Remark 8, one has
(32)∥u∥LpℝN,ℂ=∥∣u∣∥LpℝN,ℝ≤c~∥∣u∣∥s·=c~∫ℝNu2dx+∫ℝN∫ℝNux∣−∣uy2x−yN+2sx,ydxdy12≤c~∫ℝNu2dx+∫ℝN∫ℝNux−eix−y·Ax+y/2uy2x−yN+2sx,ydxdy12=c~∥u∥s·,A.
From the above inequality, we immediately obtain the embeddingDAs·ℝN,ℂ↪LpℝN,ℂ which is continuous.Forλ>0, define
(33)u,vλ≔R∫ℝ2Nux−eix−y·Ax+y/2uyvx−eix−y·Ax+y/2vy¯x−yN+2sx,ydxdy+Rλ∫ℝNV+uv¯dx,∥u∥λ≔u,uλ12.SetE=u∈DAs·ℝN,ℂ:∫ℝNV+u2dx<∞ be equipped with the inner product u,vE=u,v1(i.e., λ=1 in u,vλ). Obviously, ∥u∥E≤∥u∥λ for λ≥1. Set Xλ=X,∥·∥λ. Combining condition V4 and fractional Sobolev inequality, we could get
(34)∫ℝNux2dx=∫V+<kux2dx+∫V+≥kux2dx≤∥u∥∞2∣V+<k∣+1k∫ℝNV+xux2dx≤M2∥u∥s·,A2∣V+<k∣+1k∫ℝNV+u2dx=M2∣V+<k∣∫ℝNux2dx+us·,A2+1k∫ℝNV+u2dx,which shows that
(35)∫ℝNux2dx≤11−M2∣V+<k∣M2∣V+<k∣us·,A2+1k∫ℝNV+u2dx≤maxM2∣V+<k∣,1/k1−M2∣V+<k∣us·,A2+∫ℝNV+u2dx=maxM2∣V+<k∣,1/k1−M2∣V+<k∣∥u∥X2.From the above inequality, it holds that(36)us·,A2+∫ℝNux2dx≤1+maxM2∣V+<k∣,1/k1−M2∣V+<k∣∥u∥X2,which shows that X is continuously embedded into DAs·ℝN,ℂ. Similarly, for all λ≥1/kM2∣V+<k∣, there holds
(37)∫ℝNux2dx≤11−M2∣V+<k∣M2∣V+<k∣us·,A2+λM2∣V+<k∣∫ℝNV+u2dx=M2∣V+<k∣1−M2∣V+<k∣us·,A2+λ∫ℝNV+u2dx=M2∣V+<k∣1−M2∣V+<k∣∥u∥λ2=1θ∥u∥λ2,where θ=1−M2V+<k/M2V+<k. In addition, we have
(38)∫ℝNuxpxdx=∫ℝNuxpx−2·ux2dx≤max∥u∥∞p+−2,∥u∥∞p−−2∫ℝNux2dx≤maxMp+−2∫ℝNux2dx+us·,A2p+−22,Mp−−2∫ℝNux2dx+us·,A2p−−22M2∣V+<k∣1−M2∣V+<k∣∥u∥λ2.This togetherM2∣V+<k∣<1 yields that
(39)∫ℝNuxpxdx≤maxMp+−2M2∣V+<k∣1−M2∣V+<k∣∥u∥λ2+∥u∥λ2p+−22,Mp−−2M2∣V+<k∣1−M2∣V+<k∣∥u∥λ2+∥u∥λ2p−−22M2∣V+<k∣1−M2∣V+<k∣∥u∥λ2≤∣V+<k∣11−M2∣V+<k∣p+2maxMp+∥u∥λp+,Mp−∥u∥λp−=1θp+/2Mp+V+<kp+−2/2maxMp+∥u∥λp+,Mp−∥u∥λp−=1θp+/2V+<kp+−2/2max∥u∥λp+,Mp−−p+∥u∥λp−.For the sake of notational simplicity, we let∥u∥λ,V2≔us·,A2+∫ℝNVλu2dx. Hence, by condition V4, we have
(40)∥u∥λ2≥∥u∥λ,V2≥ϑ0−1ϑ0∥u∥λ2,forallλ≥0.Related to equation (1), we think the functional Ψλ:Xλ→ℝ,(41)Ψλu=12∥u∥λ2−12∫ℝNV−u2dx−∫ℝNfxqxuqx+gxpxupxdx=12∥u∥λ,V2−∫ℝNfxqxuqx+gxpxupxdx.In fact, we can easily verify thatΨλ is well-defined of class C1 in Xλ and
(42)Ψλ′u,v=u,vλ−R∫ℝNV−uv¯dx−R∫ℝNfxuqx−2u+gxupx−2uv¯dx=u,vλ,V−R∫ℝNfxuqx−2u+gxupx−2uv¯dx,for all u,v∈Xλ. Therefore, if u∈Xλ is a critical point of Ψλ, then u is a solution of equation (1). Since the energy functional Ψλ is unbounded below on Xλ, in order to overcome this problem, we use the Nehari manifold Nλ=u∈Xλ\0:Ψλ′u,u=0 to study the energy functional. In addition, we also note that Nλ contains every nonzero solution of equation (1). Especially, all critical points of must be located in Nλ, and the local minimizers on Nλ are usually critical points of Ψλ.


## 3. Main Results

To start with, we can get an estimate ofΨλ. Then, we will discuss some basic properties of Nλ. Finally, we prove Theorem 2 and Theorem 3 using the variational methods.Lemma 10.
Ψλ is coercive and bounded below on Nλ. Furthermore, one has
(43)Ψλu≥max−2−q+2p−ϑ0q+θϑ0−1p−−2q+2−q+p−−q−∥f∥L2/2−qxq−22−q+,−2−q−2p−q−ϑ0θϑ0−1p−−2q−2−q−p−−q−∥f∥L22−qx22−q−.Proof.
Ifu∈Nλ, in view of (37), (40), and Ho¨lder inequality, it gains
(44)Ψλu=Ψλu−1p−Ψλ′u,u=12−1p−∥u∥λ,V2−∫ℝN1qx−1p−fxuqxdx−∫ℝN1px−1p−gxupxdx≥12−1p−∥u∥λ,V2−1q−−1p−∫ℝNfxuqxdx≥12−1p−∥u∥λ,V2−1q−−1p−∥f∥L22−qxℝN∫ℝNu2dxqx2≥12−1p−∥u∥λ,V2−1q−−1p−∥f∥L22−qxℝN1θqx/2∥u∥λqx≥p−−22p−∥u∥λ,V2−1q−−1p−∥f∥L22−qxℝNmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−≥p−−22p−ϑ0−1ϑ0∥u∥λ2−1q−−1p−∥f∥L22−qxℝNmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−≥max−2−q+2p−ϑ0q+θϑ0−1p−−2q+2−q+p−−q−∥f∥L2/2−qxℝNq−22−q+,−2−q−2p−q−ϑ0θϑ0−1p−−2q−2−q−p−−q−∥f∥L22−qxℝN22−q−.
Therefore,Ψλ is coercive and bounded below on Nλ.We know thatNλ is linked to the behavior of the function of the form Lut:t→Ψλtu for t>0. This map is called as the fibering map which can be traced back to basic works [1, 22, 23]. If u∈Xλ, then
(45)Lut=t22∥u∥λ,V2−∫ℝNfxqxtuqxdx−∫ℝNgxpxtupxdx,Lu′t=t∥u∥λ,V2−∫ℝNfxtqx−1uqxdx−∫ℝNgxtpx−1upxdx;Lu′′t=∥u∥λ,V2−∫ℝNfxqx−1tqx−2uqxdx−∫ℝNgxpx−1tpx−2upxdx.After observation, we can get that(46)tLu′t=∥tu∥λ,V2−∫ℝNfxtuqxdx−∫ℝNgxtupxdxand thus, for u∈Xλ\0 and t>0,Lu′t=0 if and only if tu∈Nλ, i.e., positive critical points of Lu correspond points on the Nehari manifold. Especially, Lu′1=0 if and only if u∈Nλ. We found that Nλ can be divided into three parts corresponding local minimal, local maximum, and points of inflection. Based on the above, we can define
(47)Nλ+=u∈Nλ:Lu′′1>0;Nλ0=u∈Nλ:Lu′′1=0;Nλ−=u∈Nλ:Lu′′1<0.For eachu∈Nλ, we can find that
(48)Lu′′1=∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx.Now, we will deduce some results ofNλ+,Nλ0, and Nλ−.Lemma 11.
Assumeu0 is a local minimizer of Ψλ on Nλ and u0∉Nλ0, then Ψλ′u0=0 in Xλ−1.Proof.
Ifu0 is a local minimizer of Ψλ on Nλ, then u0 is a solution of the optimization problem
(49)minimizerΨλusubjecttoKu=0,where Ku=∥u∥λ,V2−∫ℝNfxuqxdx−∫ℝNgxupxdx. Consequently, by the theory of Lagrange multipliers, there exists ν∈ℝ such that Ψλ′u0=νK′u0. Therefore,
(50)Ψλ′u0,u0=νK′u0,u0.
It follows fromu0∈Nλ that
(51)∥u0∥λ,V2−∫ℝNfxu0qxdx−∫ℝNgxu0pxdx=0.
Thus,(52)K′u0,u0=2∥u0∥λ,V2−∫ℝNqxfxu0qxdx−∫ℝNpxgxu0pxdx=∥u0∥λ,V2−∫ℝNqx−1fxu0qxdx−∫ℝNpx−1gxu0pxdx.
Ifu0∉Nλ0, then K′u0,u0≠0. In view of (50), it gains ν=0.Lemma 12.
(1)
∀u∈Nλ+∪Nλ0, one has ∫ℝNfxuqxdx>0(2)
∀u∈Nλ−, one has ∫ℝNgxupxdx>0Proof.
By the definitions ofNλ+ and Nλ0, we can obtain
(53)0≤Lu′′1=∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx=∥u∥λ,V2−∫ℝNq−−1fxuqxdx−∫ℝNp−−1gxupxdx<2−p−∥u∥λ,V2−q−−p−∫ℝNfxuqxdx.
It is easy to get that∫ℝNfxuqxdx>p−−2/p−−q−∥u∥λ,V2≥0. It follows from the definition of Nλ− that
(54)0>Lu′′1=∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx≥∥u∥λ,V2−∫ℝNq+−1fxuqxdx−∫ℝNp+−1gxupxdx=2−q+∥u∥λ,V2−p+−q+∫ℝNgxupxdx,which implies that ∫ℝNgxupxdx>2−q+/p+−q+∥u∥λ,V2≥0.Lemma 13.
Let the condition (H3), (H4), and (V1)-(V5) are satisfied. Then, for all λ≥1/kM2∣V+<k∣, one has Nλ0=∅.Proof.
If the conclusion does not hold, then there existsλ≥1/kM2∣V+<k∣, such that Nλ0≠∅. Then, for u∈Nλ0, by (40), (48), and the Ho¨lder inequality, we have
(55)0=Lu′′1=∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx<2−p−∥u∥λ,V2−q−−p−∫ℝNfxuqxdx.This means that
(56)q−2ϑ0−1ϑ0∥u∥λ2≤ϑ0−1ϑ0∥u∥λ2≤∥u∥λ,V2<p−−q−p−−2∫ℝNfxuqxdx<p−−q−p−−2∥f∥L22−qxℝN∫ℝNuqxdx<p−−q−p−−2∥f∥L22−qxℝNmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−.Thus, we have
(57)∥u∥λ≤min2ϑ0p−−q−∥f∥L2/2−qxℝNq−ϑ0−1p−−2θq+/212−q+,2ϑ0p−−q−∥f∥L2/2−qxℝNq−ϑ0−1p−−2θq−/212−q−.
From (48), we seem to easily get that
(58)2−q+∥u∥λ,V2−p+−q+∫ℝNgxupxdx≤∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx=Lu′′1=0,which implies that
(59)2−q+p+−q+∥u∥λ,V2≤∫ℝNgxupxdx.
Combining (39) and (40) with the Sobolev inequality, we have
(60)ϑ0−12−q+ϑ0p+−q+∥u∥λ2≤2−q+p+−q+∥u∥λ,V2≤∫ℝNgxupxdx≤∥g∥∞θp+/2V+<kp+−2/2max∥u∥λp+,Mp−−p+∥u∥λp−.This means that
(61)∥u∥λ≥maxϑ0−12−q+θ∧p∧+/2∣V+<k∣∧p∧+−2/2ϑ0p+−q+∥g∥∞1p+−2,ϑ0−12−q+θ∧p∧+/2∣V+<k∣∧p∧+−2/2ϑ0p+−q+∥g∥∞Mp−−p+1p−−2.
Hence, combining (57) and (61), we have
(62)∣V+<k∣<minA1,A2,A3,A4≤maxA1,A2,A3,A4,which is a contradictive with (V5). Therefore, for all λ≥1/kM2∣V+<k∣, one has Nλ0=∅.By Lemma13, λ≥1/kM2∣V+<k∣, we can easily get that Nλ=Nλ+∪Nλ− and define
(63)cλ+=infu∈Nλ+Ψλuandcλ−=infu∈Nλ−Ψλu.Furthermore, we derive the following results.Lemma 14.
Under the condition (H3), (H4), and (V1)-(V5). Then, for all λ≥1/kM2∣V+<k∣, there exists C5 such that cλ+<0<C5<cλ−. Particularly, cλ+=infu∈NλΨλu.Proof.
Our proof is decoupled in the following two steps:Step 1.
We claim that there existu∈Nλ+ such that Ψλu<0. Indeed, let tv0∈Nλ+⊂Nλ, where t∈0,1 small enough. It is follows from (48) that
(64)0<Ltv0′′1=∥tv0∥λ,V2−∫ℝNqx−1fxtv0qxdx−∫ℝNpx−1gxtv0pxdx<2−p−∥tv0∥λ,V2−q−−p−∫ℝNfxtv0qxdx.This shows that
(65)0≤p−−2p−−q−∥tv0∥λ,V2<∫ℝNfxtv0∣qxdx<tq−∫ℝNfxv0qxdx.This yields at once that
(66)∫ℝNfxv0qxdx>0.
Similarly,(67)0<Ltv0′′1=∥tv0∥λ,V2−∫ℝNqx−1fxtv0qxdx−∫ℝNpx−1gxtv0pxdx<2−q−∥tv0∥λ,V2−p−−q−∫ℝNgxtv0pxdx.
From (67), we can easily get that
(68)2−q−p−−q−∥tv0∥λ,V2>∫ℝNgxtv0pxdx.
Consequently, it derives from (67) and (68) that
(69)Ψλtv0=Ψλtv0−1p−Ψλ′tv0,tv0<12∥tv0∥λ,V2−1q+∫ℝNfxtv0qxdx−1p+∫ℝNgxtv0pxdx−1p−∥tv0∥λ,V2+1p−∫ℝNfxtv0qxdx+1p−∫ℝNgxtv0pxdx=12−1p−∥tv0∥λ,V2−∫ℝN1q+−1p−fxtv0qxdx+1p−−1p+∫ℝNgxtv0pxdx<12−1p−∥tv0∥λ,V2−∫ℝN1q+−1p−fxtv0qxdx+1p−−1p+2−q−p−−q−∥tv0∥λ,V2=12−1p−+1p−−1p+2−q−p−−q−∥tv0∥λ,V2−∫ℝN1q+−1p−fxtv0qxdx<t212−1p−+1p−−1p+2−q−p−−q−∥v0∥λ,V2−tq+∫ℝN1q+−1p−fxv0qxdx.
Hence,cλ+<0.Step 2.
We assert that there existu∈Nλ− such that Ψλu>0. In fact, let u∈Nλ−⊂Nλ. From (48), we seem to easily get that
(70)2−q+∥u∥λ,V2−p+−q+∫ℝNgxupxdx≤∥u∥λ,V2−∫ℝNqx−1fxuqxdx−∫ℝNpx−1gxupxdx=Lu′′1<0,which implies that
(71)2−q+p+−q+∥u∥λ,V2<∫ℝNgxupxdx.
Combining (39), (40), and (71) with Sobolev inequality, we have
(72)ϑ0−12−q+ϑ0p+−q+∥u∥λ2≤2−q+p+−q+∥u∥λ,V2<∫ℝNgxupxdx≤∥g∥∞θp+/2V+<kp+−2/2max∥u∥λp+,Mp−−p+∥u∥λp−and so
(73)∥u∥λ≥maxϑ0−12−q+θ∧p∧+/2∣V+<k∣∧p∧+−2/2ϑ0p+−q+∥g∥∞1p+−2,ϑ0−12−q+θ∧p∧+/2∣V+<k∣∧p∧+−2/2ϑ0p+−q+∥g∥∞Mp−−p+1p−−2≔C6.
It follows from (44) that
(74)Ψλu≥p−−22p−ϑ0−1ϑ0∥u∥λ2−1q−−1p−∥f∥L22−qxℝNmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−>maxC6q+p−−22p−ϑ0−1ϑ0C62−q+−1q−−1p−∥f∥L22−qxℝN1θq+/2,C6q−p−−22p−ϑ0−1ϑ0C62−q−−1q−−1p−∥f∥L22−qxℝN1θq−/2.
Consequently, ifλ>1/kM2∣V+<k∣, then cλ−>C5 for some C5>0.We note that iff,g and Vλ satisfy the hypotheses in Theorem 3, we can choose φ∈C0∞Ωg,ℂ, such that Lφt=Ψλtφ=t2/2∥φ∥λ,V2−∫Ωgfx/qxtφqxdx−∫Ωggx/pxtφpxdx have t0>0 and C0 which are independent of λ that satisfy t0φ∈Nλ− for all λ>λ∗ and
(75)supt≥0Lφt=Lφt0=C0>0,which shows cλ−≤C0 for all λ>λ∗.Lemma 15.
Assume that the conditions (H1)-(H5) and (V1)-(V5) hold, then there exists λ∗≥1/kM2∣V+<k∣ such that Ψλ satisfies the PSc condition in Xλ for all c<C0 and λ>λ∗.Proof.
First, we assumeun is a PSc sequence with c<C0. In view of Lemma 10, there exists a positive constant C^ related to λ such that ∥un∥λ≤C^. Consequently, there is a subsequence which is still denote as un and u0 in Xλ such that
(76)un⇀u0inXλ,un→u0inLlocrℝN,ℂ,for2≤r≤∞,gxunpx−2un⇀gxu0px−2u0inLp′xℝN,ℂ.
Besides,Ψλ′u0=0. Let vn=un−u0. Making use of the Vitali theorem, it holds that
(77)limn→∞∫ℝNfxvnqxdx=0.
In fact, note thatf∈L2/2−qxℝN,ℂ, for any 0<ε<1; then, there exists rε>0 such that for ζ∈ℝN and r>rε,
(78)∫ℝN\Brζfx22−qxdx<ε22−q+.
For eachΩ0⊂Brζ, one has
(79)∫Ω0fxvnqxdx≤max∫Ω0fx22−qxdx2−q−2,∫Ω0fx22−qxdx2−q+2max∫Ω0vn2dxq−2,∫Ω0vn2dxq+2≤C^max∫Ω0fx22−qxdx2−q−2,∫Ω0fx22−qxdx2−q+2.
It is easy to get thatfxvnqx is a equi-integrable on Brζ. Besides, fxvnqx→0, a.e., in Brζ. It follows from the Vitali theorem that
(80)limn→∞∫Brζfxvnqxdx=0.
Hence, there holds(81)∫ℝNfxvnqxdx=∫Brζfxvnqxdx+∫ℝNBrζ0fxvnqxdx≤C^ε+∫Brζfxvnqxdx,which implies that limn→∞∫ℝNfxvnqxdx=0.
Next, we assert thatun→u0 in Xλ. In fact, by (V2), we obtain
(82)∫ℝNvn2dx=∫V+≥kvn2dx+∫V+<kvn2dx≤1λk∫V+≥kλV+vn2dx+∫V+<kvn2dx≤1λk∥vn∥λ2+o1.
In light of the Ho¨lder inequality with the Sobolev inequality, we have
(83)∫ℝNvnpxdx=∫ℝNvnpx−2·vn2dx≤max∥vn∥∞p+−2,∥vn∥∞p−−2∫ℝNvn2dx≤maxMp+−2∥vn∥s·,Ap+−2,Mp−−2∥vn∥s·,Ap−−2∫ℝNvn2dx≤maxMp+−2∥vn∥L2ℝN2+vns·,A2p+−22,Mp−−2∥vn∥L2ℝN2+vns·,A2p−−22∫ℝNvn2d≤maxMp+−21λk∥vn∥λ2+∥vn∥λ2+o1p+−22,Mp−−21λk∥vn∥λ2+∥vn∥λ2+o1p−−22∫ℝNvn2dx≤maxMp+−2λk+1λkp+−22∥vn∥λp+−2+o1,Mp−−2λk+1λkp−−22∥vn∥λp−−2+o1∫ℝNvn2dx≤1λkmaxMp+−2∥vn∥λp+−2,Mp−−2∥vn∥λp−−2λk+1λkp+−22∥vn∥λ2+o1.
By Bre′is-Lieb Lemma, we have
(84)∥un∥λ,V2=∥un−u0∥λ,V2+∥u0∥λ,V2+o1.
By applying a Bre′is-Lieb type result on variable exponent Lebesgue space (see [24]) and (H3)-(H4), it is easy to obtain that
(85)∫ℝNfxunqxdx=∫ℝNfxun−u0qxdx+∫ℝNfxu0qxdx+o1.
Similarly,(86)∫ℝNgxunpxdx=∫ℝNgxun−u0pxdx+∫ℝNgxu0pxdx+o1.
Then, overall, we can get thatΨλvn=Ψλun−Ψλu0+o1 and Ψλ′vn=o1. Then, by virtue of (77) and Lemma 10, we get that
(87)C0+C7+o1>c−Ψλu0+o1=Ψλvn−1p−Ψλ′vn,vn+o1≥p−−22p−∥vn∥λ,V2−1q−−1p−∫ℝNfxvnqxdx+o1≥p−−22p−∥vn∥λ,V2−1q−−1p−∥f∥L22−qxmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−+o1≥p−−22p−ϑ0−1ϑ0∥vn∥λ2−1q−−1p−∥f∥L22−qxmax1θq+/2∥u∥λq+,1θq−/2∥u∥λq−+o1,where
(88)C7=min2−q+2p−ϑ0q+θϑ0−1p−−2q+2−q+p−−q−∥f∥L2/2−qxq−22−q+,2−q−2p−q−ϑ0θϑ0−1p−−2q−2−q−p−−q−∥f∥L22−qx22−q−.
Suppose by contradiction thatvn is not bounded in Xλ. Then, there exists a subsequence still denoted by vn such that ∥vn∥λ→∞ as n→∞. Hence, by virtue of (87), we have
(89)C0+C7∥vn∥λ2+o11∥vn∥λ2≥p−−22p−ϑ0−1ϑ0−1q−−1p−∥f∥L22−qxmax1θq+/2∥vn∥λq+−2,1θq−/2∥vn∥λq−−2+o11∥vn∥λ2,which is contradictory since 1<q−≤q+<2<p−. Thus, vn is bounded in Xλ for all λ>λ∗≥1/kM2∣V+<k∣. That is, there exist a constant M1>0 such that ∥vn∥λ≤M1. From (83), we can get that
(90)∫ℝNvnpxdx≤1λkmaxMM1p+−2,MM1p−−2λk+1λkp+−22∥vn∥λ2+o1.
TogetherΨλ′vn,vn=o1 with (77)-(90), there holds
(91)o1=Ψλ′vn,vn=∥vn∥λ,V2−∫ℝNgxvnpxdx≥ϑ0−1ϑ0∥vn∥λ2−∥g∥∞∫ℝNvnpxdx≥ϑ0−1ϑ0∥vn∥λ2−∥g∥∞λkmaxMM1p+−2,MM1p−−2λk+1λkp+−22∥vn∥λ2+o1.
We find that there existsλ∗≥1/kM2∣V+<k∣ large enough such that
(92)∥g∥∞λkmaxMM1p+−2,MM1p−−2λk+1λkp+−22<ϑ0−1ϑ0,for all λ>λ∗. It follows from (91) that vn→0 in Xλ for all λ>λ∗.Theorem 16.
Assume that (H1)-(H5) and (V1)-(V5) hold, then there exists λ∗≥0 such that for every λ>λ∗, Ψλ has a minimizer uλ+inNλ+ satisfying that
(93)Ψλuλ+=cλ+=infu∈Nλ+Ψλu,(1)
uλ+ is a nontrivial solution of equation (1).Proof.
Combining Lemma14 and the Ekeland variational principle in [25], there exists un⊂Nλ+ such that un is a PScλ+ sequence for Ψλ. Furthermore, using Lemma 10, we can get that un is bounded in Xλ. Consequently, there exists a subsequence of un (we still denote as un) and uλ+ in Xλ such that
(94)un⇀uλ+inXλ,un→uλ+inLlocrℝN,ℂ,for2≤r≤∞.
Besides,Ψλ′uλ+=0. In view of Lemma 15, we know that un→uλ+ in Xλ and Ψλuλ+=cλ+. In other words, uλ+ is a solution of equation (1).
Now, we will check thatuλ+≠0. On the contrary, by combining (40), (H3), the Egoroff theorem and the Ho¨lder inequality, there holds ∫ℝNfxunqxdx→0 as n→∞, which shows that
(95)∥un∥λ,V2=∫ℝNgxunpxdx+o1Ψλun=12∥un∥λ,V2−∫ℝNfxqxunqxdx−∫ℝNgxpxunpxdx≥12∥un∥λ,V2−1q−∫ℝNfxunqxdx−1p−∫ℝNgxunpxdx=12∥un∥λ,V2−1p−∥un∥λ,V2+o1=p−−22p−∥un∥λ,V2+o1≥p−−2ϑ0−12p−ϑ0∥un∥λ2+o1≥0.
This is contradictive withlimn→∞Ψλun=cλ+<0. Hence, uλ+≠0; that is, uλ+ is a nontrivial solution of equation (1).Proof of Theorem 17.
The result of Theorem2 is immediately available from Theorem 20.Theorem 18.
Assume that the conditions (H3), (H4), and (V1)-(V5) are satisfied, then there exists λ∗≥0 such that for every λ>λ∗, Ψλ has a minimizer uλ−inNλ− satisfying that
(96)Ψλuλ−=cλ−,(1)
uλ− is a nontrivial solution of equation (1).Proof.
According to Lemma14 and the Ekeland variational principle in [25], there exists un⊂Nλ− such that un is a PScλ− sequence for Ψλ. Furthermore, using Lemma 10, we can get that un is bounded in Xλ. Consequently, there exists a subsequence of un (we still denote as un) and uλ− in Xλ such that
(97)un⇀uλ−inXλ,un→uλ−inLlocrℝN,for2≤r≤∞.
Besides,Ψλ′uλ−=0. In view of Lemma 15, we know that un→uλ− in Xλ and Ψλuλ−=cλ−. In other words, uλ− is a solution of equation (1).
Now, we will check thatuλ−≠0. Suppose the contrary, combining (39), (40), Egoroff theorem, and (H4), there holds ∫ℝNgxunpxdx→0 as n→∞, which shows that
(98)unλ,V2=∫ℝNfxunqxdx+o1Ψλun=12unλ,V2−∫ℝNfxqxunqxdx−∫ℝNgxpxunpxdx≤12unλ,V2−1q+∫ℝNfxunqxdx−1p+∫ℝNgxunpxdx=12unλ,V2−1q+unλ,V2+o1=q+−22q+unλ,V2+o1≤q+−2ϑ0−12q+ϑ0unλ2+o1≤0.
This is contradictive withlimn→∞Ψλun=cλ−>0. Hence, uλ−≠0; that is, uλ− is a nontrivial solution of equation (1).Proof of Theorem 19.
It derives from Theorem20, Theorem 22, and Lemma 14 that equation (1) has two nontrivial solutions uλ+ and uλ− such that uλ+∈Nλ+ and uλ−∈Nλ− with Ψλuλ+=cλ+<0<C5<Ψλuλ−=cλ−.Theorem 20.
Assume that (H3), (H4), and (V1)-(V5) hold, then for all λ≥1/kM2∣V+<k∣, Ψλ has a minimizer uλ+inNλ+ satisfying that
(99)Ψλuλ+=cλ+=infu∈Nλ+Ψλu;(1)
uλ+ is a nontrivial solution of equation (1).Proof.
Combining Lemma14 and the Ekeland variational principle in [25], there exists un⊂Nλ+ such that un is a PScλ+ sequence for Ψλ. Furthermore, using Lemma 10, we can get that un is bounded in Xλ. Consequently, there exists a subsequence of un (we still denote as un) and uλ+ in Xλ such that
(100)un⇀uλ+inXλ,un→uλ+inLlocrℝN,for2≤r≤∞.
Besides,Ψλ′uλ+=0. In view of Lemma 15, we know that un→uλ+ in Xλ and Ψλuλ+=cλ+. In other words, uλ+ is a solution of equation (1).
Now, we will check thatuλ+≠0. On the contrary, by combining (40), (H3), the Egoroff theorem, and the Ho¨lder inequality, there holds ∫ℝNfxunqxdx→0 as n→∞, which shows that
(101)∥un∥λ,V2=∫ℝNgxunpxdx+o1Ψλun=12∥un∥λ,V2−∫ℝNfxqxunqxdx−∫ℝNgxpxunpxdx≥12∥un∥λ,V2−1q−∫ℝNfxunqxdx−1p−∫ℝNgxunpxdx=12∥un∥λ,V2−1p−∥un∥λ,V2+o1=p−−22p−∥un∥λ,V2+o1≥p−−2ϑ0−12p−ϑ0∥un∥λ2+o1≥0.
This is contradictive withlimn→∞Ψλun=cλ+<0. Hence, uλ+≠0; that is, uλ+ is a nontrivial solution of equation (1).Proof of Theorem 21.
The result of Theorem2 is immediately available from Theorem 20.
We note that iff,g and Vλ satisfy the hypotheses in Theorem 3, we can choose φ∈C0∞Ωg, such that Lφt=Ψλtφ=t2/2∥φ∥λ,V2−∫Ωgfx/qxtφqxdx−∫Ωggx/pxtφpxdx have t0>0 and C8 which are independent of λ that satisfy t0φ∈Nλ− for all λ>λ∗ and
(102)supt≥0Lφt=Lφt0=C8>0,which shows cλ−≤C8 for all λ>λ∗.Theorem 22.
Assume that the conditions (H3), (H4), and (V1)-(V5) are satisfied, then for all λ>1/kM2∣V+<k∣, Ψλ has a minimizer uλ−inNλ− satisfying that
(103)Ψλuλ−=cλ−;(1)
uλ− is a nontrivial solution of equation (1).Proof.
According to Lemma14 and the Ekeland variational principle in [25], there exists un⊂Nλ− such that un is a PScλ− sequence for Ψλ. Furthermore, using Lemma 10, we can get that un is bounded in Xλ. Consequently, there exists a subsequence of un (we still denote as un) and uλ− in Xλ such that
(104)un⇀uλ−inXλ,un→uλ−inLlocrℝN,for2≤r≤∞.
Besides,Ψλ′uλ−=0. In view of Lemma 15, we know that un→uλ− in Xλ and Ψλuλ−=cλ−. In other words, uλ− is a solution of equation (1).
Now, we will check thatuλ−≠0. Suppose the contrary, combining (39), (40), Egoroff theorem, and (H4), there holds ∫ℝNgxunpxdx→0 as n→∞, which shows that
(105)unλ,V2=∫ℝNfxunqxdx+o1Ψλun=12unλ,V2−∫ℝNfxqxunqxdx−∫ℝNgxpxunpxdx≤12unλ,V2−1q+∫ℝNfxunqxdx−1p+∫ℝNgxunpxdx=12unλ,V2−1q+unλ,V2+o1=q+−22q+unλ,V2+o1≤q+−2ϑ0−12q+ϑ0unλ,V2+o1≤0.
This is contradictive withlimn→∞Ψλun=cλ−>0. Hence, uλ−≠0; that is, uλ− is a nontrivial solution of equation (1).Proof of Theorem 23.
It derives from Theorem20, Theorem 22, and Lemma 14 that equation (1) has two nontrivial solutions uλ+ and uλ− such that uλ+∈Nλ+ and uλ−∈Nλ− with Ψλuλ+=cλ+<0<C5<Ψλuλ−=cλ−.



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*Source: 1320635-2020-10-28.xml*