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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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On a class of nonlocal nonlinear Schrödinger equations with potential well

Tsung-fang Wu
Published Online: 2019-07-20 | DOI: https://doi.org/10.1515/anona-2020-0020

Abstract

In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.

Keywords: Nonlocal nonlinear Schrödinger equations; Nehari manifold; Multiple positive solutions; Concentration-compactness principle

MSC 2010: 35B38; 35B40; 35J20; 35J61

1 Introduction

In this paper we are concerned with the existence and multiplicity of positive solutions of the nonlocal nonlinear Schrödinger equation

Δu+Vμ,λxu+Iαupup2u=f(x)u2p2u in RN,uH1RN,(Pμ,λ)

where N3,N+αNp<NN2 and Iα is the Riesz potential of order 0 < α < min {N, 2} (2=2NN2) on the Euclidean space ℝN, defined for each point x ∈ ℝN∖{0} by

Iαx=Γ(Nα2)Γ(α2)πN/22αxNα

with Γ being the Euler gamma function. Throughout this paper, we assume that the parameters μ, λ > 0 and the functions Vμ,λ := μgλ a and f satisfy the following conditions:

  • (V1)

    g is a nonnegative continuous function on ℝN;

  • (V2)

    there exists c > 0 such that the set {g < c} := {x ∈ ℝNg(x) < c} is nonempty and has finite measure;

  • (V3)

    Ω = int{x ∈ ℝNg(x) = 0} is nonempty bounded domain and has a smooth boundary with Ω = {x ∈ ℝNg(x) = 0};

  • (V4)

    aLN/2(ℝN) ∩ L(ℝN) and ∣ {xΩ:a(x) > 0} ∣ > 0;

  • (F1)

    fL(ℝN) and ∣ {xΩ:f(x) > 0} ∣ > 0.

Remark 1.1

By condition (V4), the set {xΩ : a(x) > 0} has positive Lebesgue measure, we can assume that λ1 (aΩ) denote the positive principal eigenvalue of the problem

Δu(x)=λa¯Ω(x)u(x)forxΩ;u(x)=0forxΩ,(1.1)

where aΩ is a restriction of a on Ω Clearly, λ1 (aΩ) has a corresponding positive principal eigenfunction ϕ1.

In recent years, nonlinear Schrödinger type equation has been widely studied under variant assumptions on potential g and weight function f. Most of the literature has focused on the equation for g being a positive potential and f being a positive weight function with satisfies the some assumptions of infinite limits. Moreover, the conditions (V1) − (V3) imply that μ g represents a potential well whose depth is controlled by μ. μ g is called a steep potential well if μ is sufficiently large and one expects to find solutions which localize near its bottom Ω. This problem has found much interest after being first introduced by Bartch and Wang [11] in the study of the existence of positive solutions for nonlinear Schrödinger equations and has been attracting much attention, see [3, 9, 10, 38, 43] and the references therein. Later, the steep potential well is introduced to the study of some other types of nonlinear differential equations by some researchers, such as nonlocal nonlinear elliptic equations [18, 24, 33, 34, 45, 46].

When N = 3 and the nonlocal nonlinear term (Iαup) ∣up−2 u = (Iu2) u for α = p = 2. Then Eq. (Pμ,λ) is the one type of the following nonlocal nonlinear Schrödinger equation:

Δu+Vxu+σ(Iu2)u=f(x)u2p2u in R3,(1.2)

where 1 < p < 3 and the parameter σ > 0. It is easy to know that u is a solution of Eq. (1.2) if and only if (u, ϕ) is a solution of the following equation:

u+V(x)u+σϕu=f(x)u2p2u,in R3,ϕ=u2,in R3.(1.3)

It is well known that Eq. (1.3) is called the Schrödinger–Poisson system, which was first introduced in [7] as a physical model describing a charged wave interacting with its own electrostatic field. Eq. (1.3) also appears in the electromagnetic field, semiconductor theory, nonlinear optics and plasma physics. Due to the important applications in physics, Eq. (1.3) has been widely studied via modern variational methods under various hypotheses on the potential function and the nonlinearity; see [2, 4, 16, 23, 30, 31, 35, 36, 37] and the references therein. More precisely, Ruiz [30] obtained the existence, nonexistence and multiplicity of radial positive solutions for Eq. (1.3) with V = f ≡ 1. It turn out that p = 32 is a critical value for the existence of nontrivial solutions. Ruiz’s approach is based on minimizing the energy functional I associated with Eq. (1.3) on a certain manifold that is the Nehari–Pohozaev manifold:

Nr={uHr1(R3){0}:Q(u)=0},

where Hr1(ℝ3) consists of radially symmetric functions in H1(ℝ3) and Q(u) = 0 is derived by subtracting the Pohozaev identity of equations (1.3) from the equation 2〈I(u), u〉 = 0. They proved that when 1 < p < 3/2 and for σ is sufficiently small, Eq. (1.3) has two positive radial solutions v1, v2 with

0<Iv1<Iv2=infuHr1(R3)Iu<0.

In recent years, many authors have been studying such topics (existence of two positive solutions which one of the negative energy), for example, Chen [14], Huang et al. [21, 22] and Shen and Han [32], consider the following Schrödinger–Poisson system

u+1λhxu+lxϕu=f(x)u2p2u,in R3,ϕ=lxu2,in R3,(1.4)

where 2 < p < 3, l ∈ L2(ℝ3), fC(ℝ3) changes sign in ℝ3 and limx∣→∞f(x) = f < 0. They proved that system (1.4) has two positive solutions which one of the negative energy for λ > λ1 (h), where λ1 (h) is the first eigenvalue of −Δ + id in H1(ℝ3) with nonnegative weight function hL3/2(ℝ3).

Very recently, it has proven in [36] that the problem (1.3) admits a positive solution when V ≡ 1,1 < p ≤ 2, and σ belongs to a certain interval. To this end, the authors introduced the filtration of the Nehari manifold 𝓝, that is

N(c)={uN:I(u)<c},

and showed that this set 𝓝(c) under the given assumptions is the union of two disjoint nonempty sets, namely,

N(c)=N1N2,

which are both C1 sub-manifolds of 𝓝(c) and natural constraints of I. Moreover, 𝓝(1) is bounded such that I is coercive and bounded below on it, whereas I is unbounded below on 𝓝(2). Moreover, they use the argument of concentration compactness principle to obtain a minimizer of I on #x1d4dd;(1), which is a critical point of I. Actually the authors also established 𝓝(2) may not contain any non-zero critical point of I for 1+736 < p ≤ 2.

Motivated by the above works [14, 21, 22, 30, 32, 36], in the present article we mainly study the existence and multiplicity of positive solutions for Eq. (Pμ,λ) can not require conditions f changes sign in ℝ3 and limx∣→∞f(x) = f < 0. Furthermore, the existence of least energy positive solutions with negative energy and asymptotic behavior of positive solutions are also discussed. The main method of this paper is to consider minimization on two distinct components of the Nehari manifold corresponding to Eq. (Pμ,λ). The approach to Eq. (Pμ,λ) has been inspired by the papers of [12, 13, 44]. They used the Nehari manifold and fibrering maps to study the bifurcation phenomena for a nonlinear elliptic problem on bounded domains or ℝN. Since Eq. (Pμ,λ) is on ℝN, its variational setting is characterized by lack of compactness. To overcome this difficulty we apply a simplified version of the steep well method of [11] and concentration compactness principle of [27]. Furthermore, the first eigenvalue of problem −Δu + μ g(x) u = λ a(x) u in ℝN is less than λ1 (aΩ), which indicates that the original method at [12, 13, 21] cannot be directly applied, thus we provide an approximation estimate of eigenvalue to prove that the existence of positive solution for Eq. (Pμ,λ) when 0 < λ < λ1 (aΩ).

The first result is to establish the existence of least energy positive solutions and the asymptotic behavior of the solutions for Eq. (Pμ,λ) with 0 < λ < λ1 (aΩ).

Theorem 1.1

For any 0 < λ < λ1 (aΩ) there exists μ͠0(λ) > 0 with limλλ1a¯Ω μ͠0(λ) = ∞ such that for every μ > μ͠0(λ), Eq. (Pμ,λ) has a least energy positive solution uμ,λ.

Next, we now consider what happens as λλ1 (aΩ) or μ → ∞. Let

Bu:=RNIαupupdx+RNfu2pdx.

Then we have the following result.

Theorem 1.2

  1. Suppose that B(ϕ1) > 0. Let λnλ1 (aΩ) and μn > μ͠0(λn) be as in Theorem 1.1 and let un := uμn,λn be the least energy positive solution of Eq. (Pμn,λn) obtained by Theorem 1.1. Then un → 0 in X as n → ∞.

  2. For 0 < λ < λ1 (aΩ). Let uμ,λ be the least energy positive solution obtained in Theorem 1.1. Then uμ,λuλ in X as μ → ∞, where uλH01(Ω) is a positive solution of

    Δuλa¯Ωxu+Iαupup2u=fxu2p2uinΩ,u=0,onΩ.(P∞)

    The second result is to establish the existence of multiple positive solutions for Eq. (Pμ,λ) with λ > λ1 (aΩ).

Theorem 1.3

Suppose that B(ϕ1) < 0. Then there exists δ0 > 0 such that for any λ1 (aΩ) < λ < λ1 (aΩ) + δ0 and for μ enough large, Eq. (Pμ,λ) has two positive solutions uμ,λ(1) and uμ,λ(2) with uμ,λ(1) is negative energy and uμ,λ(2) is positive energy. Furthermore, uμ,λ(1) is the least energy positive solution of Eq. (Pμ,λ).

Finally, we investigate the nature of least energy positive solution uμ,λ(1) as λλ1+ (aΩ) and μ → ∞. As mentioned in the introduction a curve of positive solutions bifurcates to the right at λ1 (aΩ) when B(ϕ1) < 0 and μ sufficiently large. The following theorem implies that uμ,λ(1) will lie on this branch and the concentration of of the solutions for Eq. (Pμ,λ) with λ > λ1 (aΩ).

Theorem 1.4

  1. Suppose that B(ϕ1) < 0. Let λnλ1+ (aΩ) and μn → ∞ be as in Theorem 1.3 and let un1:=uμn,λn1 be the least energy positive solutions of Eq. (Pμn,λn) obtained by Theorem 1.3. Then

    un0;ununμnϕ1inXasn.

  2. For λ1 (aΩ) < λ < λ1 (aΩ) + δ0. Let uμ,λj (j = 1, 2) be the positive solutions obtained in Theorem 1.3. Then uμ,λjuλj, in X as μ → ∞, where uλj,H01(Ω) are positive solutions of Eq. (P).

Remark 1.2

In fact, our method can also be applied to the Choquard equation involving nonautonomous perturbation:

Δu+Vμ,λxu=Iαupup2u+f(x)u2p2uinRN,uH1RN,(Cμ,λ)

and obtain the same conclusions as all the previous theorems under the same hypotheses and in addition f is change sign in Ω. Since the proofs are similarly, and so we leave it to the reader to check. Some progress on the existence of positive solutions to Eq. (Cμ,λ), can be refer to [40, 41].

The plan of the paper is as follows. In Section 2, some preliminary results are presented and we discuss the Nehari manifold and examine carefully the connection between the Nehari manifold and the fibrering maps. In Section 3, we discuss the Nehari manifold when λ < λ1 (aΩ). In particular, we prove that Theorems 1.1, 1.2. In Section 4, we discuss the case when λ > λ1 (aΩ). In particular, we prove that Theorems 1.3, 1.4.

Throughout this paper we denote a strong convergence by “→” and a weak convergence by “⇀”.

2 Variational setting and Preliminaries

In this section, we give the variational setting for Eq. (Pμ,λ). Let

X=uH1RN|RNgu2dx<

be equipped with the inner product and norm

u,v=RNuv+guvdx,u=u,u1/2.

For μ > 0, we also need the following inner product and norm

u,vμ=RNuv+μguvdx,uμ=u,uμ1/2.

It is clear that ∥⋅∥ ≤ ∥⋅∥μ for μ ≥ 1 and set Xμ = (X, ∥⋅∥μ). It follows from conditions (V1) and (V2) and the Hölder and Sobolev inequalities that we have

RNu2+u2dx=RNu2dx+g<cu2dx+gcu2dx1+g<c2NS2RNu2dx+1cRNgu2dxmax1+g<c2NS2,1cRNu2dx+gu2dx,

this implies that the imbedding XH1(ℝN) is continuous, where the set {gc} := {x ∈ ℝNg(x) ≥ c}. Moreover, using conditions (V1) and (V2), and the Hölder and Sobolev inequalities again, we have for any r ∈ [2, 2],

RNurdxgcu2dx+g<cu2dx2r22S2NN2RNu2dxNN2r2221μcRNμ0gu2dx+g<c2NS2RNu2dx2r22(S2NN2uμ2NN2)r222g<c2r2Sruμr for μμ0:=S2c1g<c2N,(2.1)

where, S the best constant for the embedding of D1,2(ℝN) in L2(ℝN). Moreover, if we assume that uL2NpN+αRN, then by the Hardy–Littlewood–Sobolev inequality (see [20, 25, 26]) to the function ∣upL2NN+αRN, we obtain, in view of the Hölder inequality and (2.1),

RNIαupupdxRNIαup2NNαdxNα2NRNu2NpN+αdxN+α2NCN,α,2NN+αRNu2NpN+αdxN+αN(2.2)

CN,α,2NN+αg<c1p(N2)N+αS2NpN+αuμ2p,(2.3)

where

CN,α,2NN+α=ΓNα22απα/2ΓN+α2ΓN2ΓNα/N.

We use the variational methods to find positive solutions of Eq. (Pμ,λ). Associated with the Eq. (Pμ,λ), we consider the energy functional Jμ,λ : X → ℝN

Jμ,λu=12Aμ,λu12pBu,

where

Aμ,λu:=uμ2λRNau2dx

and

Bu:=RNIαupupdx+RNfu2pdx.

Because the energy functional Jμ,λ is not bounded below on X, it is useful to consider the functional on the Nehari manifold (see [29])

Nμ,λ=uX0|Jμ,λu,u=0.

Thus, uNμ,λ if and only if

Aμ,λuBu=0.

Hence, if uNμ,λ, then

Jμ,λ(u)=1212pAμ,λu=1212pBu.

Note that Nμ,λ contains every nonzero solution of Eq. (Pμ,λ). It is useful to understand Nμ,λ in terms of the stationary points of mappings of the form hu(t) = Jμ,λ(tu)(t > 0). Such a map is known as the fibrering map. It was introduced by Drébek and Pohozaev [17], and further discussed by Brown and Zhang [12]. It is clear that, if u is a local minimizer of Jμ,λ, then hu has a local minimum at t = 1. Moreover, tuNμ,λ if and only if hu(t) = 0 for uX ∖ {0}. Thus, points in Nμ,λ correspond to stationary points of the maps hu and so it is natural to divide Nμ,λ into three subsets Nμ,λ+,Nμ,λ and Nμ,λ0 corresponding to local minima, local maxima and points of inflexion of fibrering maps. We have

hu(t)=tAμ,λut2p1Bu(2.4)

and

hu(t)=Aμ,λu2p1t2p2Bu.

Hence if we define

Nμ,λ+=uNμ,λ:Aμ,λu2p1Bu>0;Nμ,λ0=uNμ,λ:Aμ,λu2p1Bu=0;Nμ,λ=uNμ,λ:Aμ,λu2p1Bu<0,

which indicates that for uNμ,λ, we have hu(1) = 0 and uNμ,λ+,Nμ,λ0,Nμ,λ if hu(1)>0,hu(1)=0,hu(1) < 0, respectively. Moreover, it is easy to show that

Nμ,λ+=uNμ,λ:Aμ,λu2p1Bu>0=uNμ,λ:2p2Bu<0=uNμ,λ:Bu<0.

Similarly,

Nμ,λ=uNμ,λ:Bu>0

and

Nμ,λ0=uNμ,λ:Bu=0.

Moreover, by (2.4), if Aμ,λ (u) and B(u) have the same sign, then hu has exactly one turning point at

t(u)=Aμ,λuBu12p2(2.5)

and if Aμ,λ (u) and B(u) have opposite signs, then hu has no turning points. Thus, if Aμ,λ (u), B(u) > 0, then hu(t) > 0 for t small and positive but hu(t) → −∞ as t → ∞; also hu(t) has a unique (maximum) stationary point at t(u) and t(u)uNμ,λ. Similarly, if Aμ,λ (u),B(u) < 0, hu(t) < 0 for t small and positive, hu(t) → ∞ as t → ∞ and hu(t) has a unique minimum at t(u) so that t(u)uNμ,λ+. Finally, if Aμ,λ (u) B(u) < 0, hu is strictly increasing (resp. decreasing) for all t > 0. Thus, we have the following results.

Lemma 2.1

If uX∖{0}, then

  1. a multiple of u lies is Nμ,λ if and only if Aμ,λ (u),B(u) > 0;

  2. a multiple of u lies is Nμ,λ+ if and only if Aμ,λ (u),B(u) < 0;

  3. when Aμ,λ (u) B(u) < 0, no multiple of u lies in Nμ,λ.

The following Lemma shows that minimizers on Nμ,λ are critical points for Jμ,λ in X.

Lemma 2.2

Suppose that u0 is a local minimizer for Jμ,λ on Nμ,λ and that u0 ȩ Nμ,λ0. Then Jμ,λ(u0) = 0.

Proof

The proof of Lemma 2.2 is essentially same as that in Brown and Zhang [12, Theorem 2.3] (or see Binding et al. [5]), so we omit it here.□

In order to prove main results, we will use a special case of the classical Brezis–Lieb lemma [8] for Riesz potentials.

Lemma 2.3

(BrezisLieb lemma for the Riesz potential [28, Lemma 2.4]). Let {un} be a bounded sequence in L2(ℝN). If unu a.e. inN, then

RNIαunupunupdx=RNIαunpunpdxRNIαupupdx+o1.

We need the following result.

Lemma 2.4

Let μn → ∞ as n → ∞ and {vn} ⊂ X withvnμnc0 for some c0 > 0. Then there exist subsequence {vn} and v0H01 (Ω) such that vnv0 in X and vnv0 in Lr(ℝN) for all 2 ≤ r < 2 and B(vn) → B(v0).

Proof

Since ∥ vn∥ ≤ ∥ vnμnc0. We may assume that there exists v0X such that

vnv0 in X,vnv0 a.e. in RN,vnv0 in LlocrRN for 2r<2.

By Fatou’s Lemma, we have

RNgv02dxlim infnRNgvn2dxlim infnvnμn2μn=0,

this implies that RNgv02dx=0 or v0 = 0 a.e. in ℝNΩ and v0H01(Ω) by condition (V3). We now show that vnv0 in Lr(ℝN). Suppose on the contrary. Then by Lions vanishing lemma (see [27, Lemma I.1] or [42, Lemma 1.21]), there exist d0 > 0,R0 > 0 and xn ∈ ℝN such that

BNxn,R0vnv02dxd0.

Moreover, xn → ∞, and hence, ∣ B(xn,R0) ∩ {x ∈ ℝN : g < c} ∣ → 0. By the Hölder inequality, we have

Bxn,R0g<cvnv02dx0.

Consequently,

c0vnμn2μncBxn,R0gcvn2dx=μncBxn,R0gcvnv02dx=μncBxn,R0vnv02dxBxn,R0g<cvnv02dx+o(1),

which a contradiction. Thus, vnv0 in Lr(ℝN) for all 2 ≤ r < 2. Moreover, by (2.2) and Lemma 2.3, B(vn) → B(v0), since 22NpN+α<2. This completes the proof.□

Next, we consider the following eigenvalue problem

Δu(x)+μgxux=λa(x)u(x) forxRN.(2.6)

We can approach this problem by a direct method and attempt to obtain nontrivial solutions of problem (2.6) as relative minima of the functional

Iμu=12RNu2+μgu2dx,

on the unit sphere in 𝔹 = {uX : ∫Nau2dx = 1}. Equivalently, we may seek to minimize a quotient as follows

λ~1,μa=infuX0RNu2+μgu2dxRNau2dx.(2.7)

Then, by (2.1),

RNu2+μgu2dxRNau2dxS2ag<c2N for all μμ0,

this implies that λ~1,μaS2ag<c2N>0. Moreover, by condition (V3),

infuX0RNu2+μgu2dxRNau2dxinfuH01Ω0RNu2+μgu2dxRNau2dx=infuH01Ω0Ωu2Ωa¯Ωu2dx,

which indicates that λ͠1,μ(a) ≤ λ1 (aΩ) for all μ > 0. Then we have the following results.

Lemma 2.5

For each μ > μ0 there exists a positive function φμX with RNaφμ2dx=1 such that

λ~1,μa=RNφμ2+μgφμ2dx<λ1a¯Ω.

Furthermore, λ͠1,μ(a) → λ1(aΩ) and φμϕ1 as μ → ∞, where ϕ1 is positive principal eigenfunction of problem (1.1).

Proof

Let {un} ⊂ X with RNaun2dx=1 be a minimizing sequence of (2.7), that is

RNun2+μgun2dxλ~1,μa as n.

Since λ͠1,μ(a) ≤ λ1 (aΩ) for all μ ≥ 0, there exists C0 > 0 independent of μ such that ∥ unμC0. Thus, there exist a subsequence {un} and φμX such that

unφμ in Xμ,unφμ a.e. in RN,unφμ in LlocrRN for 2r<2.

Moreover, by condition (V4),

RNaun2dxRNaφμ2dx=1.

Now we show that unφμ in Xμ. Suppose on the contrary. Then

RNφμ2+μgφμ2dx<lim infnRNun2+μgun2dx=λ~1,μa,

which is impossible. Thus, unφμ in Xμ, which implies that RNaφμ2dx=1 and RNφμ2+μgφμ2dx = λ͠1,μ(a). Since ∣ φμ∣ ∈ X and

λ~1,μa=RNφμ2+μgφμ2dx=RNφμ2+μgφμ2dx,

by the maximum principle, we may assume that φμ is positive eigenfunction of problem (Pμ). Moreover, by the Harnack inequality due to Trudinger [39], we must have λ͠1,μ(a) < λ1 (aΩ). Now, by the definition of λ͠1,μ(a), there holds λ͠1,μ1 (a) ≤ λ͠1,μ2 (a) for μ1 < μ2. Hence, for any sequence μn → ∞, let φn := φμn be the minimizer of λ1,μn (a). Then RNaφn2dx=1 and

λ~1,μna=RNφn2+μngφn2dxλ1a¯Ω,

that

λ~1,μnad0λ1a¯Ω for some d0>0

and

φnφnμnλ1a¯Ω, for n sufficiently large.

Thus, by Lemma 2.4, we may assume that there exists φ0H01(Ω) such that φnφ0 in X and φnφ0 in Lr(ℝN) for all 2 ≤ r < 2. Then

Ωφ02dxlim infnRNφn2+μngφn2dx=d0

and

limnRNaφn2dx=Ωa¯Ωφ02dx=1.

Since d0λ1 (aΩ) and λ1 (aΩ) is positive principal eigenvalue of problem (1.1). Thus, we must has ∫Ω∣ ∇ φ02 dx = λ1(aΩ) and φ0 = ϕ1 a positive principal eigenfunction of problem (1.1), which completes the proof.□

3 The Proof of Theorems 1.1, 1.2 (λ < λ1 (aΩ))

First, we investigate the behavior of Jμ,λ on Nμ,λ.

Lemma 3.1

For each 0 < λ < λ1 (aΩ) there exists μ0(λ) ≥ μ0 with limλλ1a¯Ω μ0(λ) = ∞ such that for every μ > μ0(λ), we have

  1. Nμ,λ = Nμ,λ;

  2. the energy functional Jμ,λ is coercive and bounded below on Nμ,λ. Furthermore, there exists d0 > 0 such that

    infuNμ,λJμ,λ(u)p1λ~1,μaλ2pλ1,μad01/p1>0(3.1)

    for all uNμ,λ.

Proof

  1. By Lemma 2.5, for each 0 < λ < λ1(aΩ) there exists μ0(λ) ≥ μ0 such that for every μ > μ0(λ), there holds λ < λ͠1,μ(a) ≤ λ1 (aΩ), which indicates that

    Aμ,λu=uμ2λRNau2dxλ~1,μaλλ~1,μauμ2>0 for all uX0.(3.2)

    Thus, by Lemma 2.1, the submanifolds Nμ,λ+ and Nμ,λ0 are empty and so Nμ,λ=Nμ,λ.

  2. By (2.1) and (3.2), for each μ > μ0(λ) and uNμ,λ, we obtain

    λ~1,μaλλ~1,μauμ2Aμ,λu<2p1Bu2p1fg<c22p2S2puμ2p,

    which indicates that

    uμd0:=Spλ~1,μaλ2p1λ~1,μafg<c22p21/2p2.

    Thus,

    Jμ,λ(u)=p12pAμ,λup1λ~1,μaλ2pλ~1,μad01/p1>0,

    this implies that the energy functional Jμ,λ is coercive and bounded below on Nμ,λ. This completes the proof.

    We now show that there exists a minimizer on Nμ,λ which is a critical point of Jμ,λ(u) and so a nontrivial solution of Eq. (Pμ,λ). First, we define

    cλ(Ω)=infuMμ,λ(Ω)Jμ,λ|H01(Ω)(u),

    where

    Mμ,λ(Ω)={uH01(Ω):Jμ,λ|H01(Ω)u,u=0}.

    Note that

    Jμ,λ|H01Ω(u)=12Ωu2dxΩλa¯Ωu2dx12pΩIαupupdx+Ωfu2pdx,

    a restriction of Jμ,λ on H01(Ω), and cλ(Ω) independent of μ. Since 0 < λ < λ1(aΩ), similar to the argument of (3.1), we can conclude that Jμ,λ|H01(Ω) is bounded below on Mμ,λ(Ω). Moreover, H01(Ω) ⊂ Xμ for all μ > 0, one can see that

    0<ηinfuNμ,λJμ,λ(u)cλ(Ω) for all μμ0.

    Taking D0 > cλ(Ω). Then we have

    0<ηinfuNμ,λJμ,λ(u)cλ(Ω)<D0(3.3)

    for all μμ0. Furthermore, we have the following results.

Theorem 3.2

For each 0 < λ < λ1 (aΩ) there exists μ͠0(λ) ≥ μ0(λ) such that Jμ,λ has a minimizer on Nμ,λ for all μ > μ͠0(λ).

Proof

By Lemma 3.1 and the Ekeland variational principle [19], for each μ > μ0(λ) there exists a minimizing sequence {un} ⊂ Nμ,λ such that

limnJμ,λ(un)=infuNμ,λJμ,λ(u)>0 and Jμ,λ(un)=o1.

Since infuNμ,λJμ,λ(u) < D0, again using Lemma 3.1, there exists C0 > 0 such that ∥ unμC0. Thus, there exist a subsequence {un} and u0X such that Jμ,λ(u0) ≥ 0, Jμ,λ(u0) = 0 and

unu0 in Xμ,unu0 a.e. in RN,unu0 in LlocrRN for 2r<2.(3.4)

Then by condition (V4),

limnRNaun2dx=RNau02dx.(3.5)

Moreover, follows from Brezis–Lieb lemma [8] and Lemma 2.3 obtain that

Bunu0=BunBu0+o(1).(3.6)

Now we show that unu0 in Xμ. Let vn = unu0. Then vn ⇀ 0 in Xμ. By the Sobolev and Gagliardo–Nirenberg inequalities, for any μ > μ0(λ) we have that

RNvn2dx1μcgcμgvn2dx+g<cvn2dx1μcvnμ2+o1

and

RNvn2pdxC01μcvnμ222p22RNvn2dx2p122+o1C01μc22p22vnμ2p+o1

or

RNvn2pdxΠμvnμ2p+o(1),(3.7)

where Πμ=C01μc22p22. Thus, using (3.4)(3.6) gives

Jμ,λvn=Jμ,λunJμ,λu0+o1 and Jμ,λ(vn),vn=o(1).(3.8)

Consequently, by (3.5), (3.6), (3.8) and Lemma 3.1, one has

D0infuNμ,λJμ,λ(u)Jμ,λu0Jμ,λvn12pJμ,λ(vn),vn+o1p1λ~1,μaλ2pλ~1,μavnμ2+o(1),

which shows that there exists a constant C1 > 0 such that

vnμC1+o1 for λ>μ¯0λ.(3.9)

Since 1<N+αN<p<2α, it follows from (3.5), (3.7) and (3.9) that

o1=Jμ,λ(vn),vnvnμ21fΠμvnμ2p2+o(1)vnμ21fΠμC12p2+o1.(3.10)

Notice that Πμ → 0 as μ → ∞. Then by (3.10), there exists μ͠0(λ) ≥ μ0(λ) such that for μ > μ͠0(λ), there holds vn → 0 in Xμ. Hence unu0 in Xμ and so

Jμ,λ(u0)=limnJμ,λ(un)=infuNμ,λJμ,λ(u),

which indicates that u0 is a minimizer on Nμ,λ. This completes the proof.□

We are now ready to prove Theorem 1.1: By Theorem 3.2, Jμ,λ has a minimizer u0 on Nμ,λ for all μ > μ͠0(λ). Since B(u0) > 0 and u0Nμ,λ0, by Lemma 2.2, u0 is a critical point of Jμ,λ. Since Jμ,λ(|u|) = Jμ,λ(u), then without loss of generality we may assume that u0 is positive. This completes the proof.

Next, we now consider what happens as λλ1a¯Ω or μ → ∞. As might be expected from the introduction the sign of B(ϕ1) plays an important role. We conclude the following results by considering the case where B(ϕ 1) > 0.

Theorem 3.3

Suppose that B(ϕ1) > 0. Then

limλλ1a¯ΩinfuNμ,λJμ,λ(u)=0.

Proof

We may assume without loss of generality that ∥ϕ1μ = 1. For 0 < λ < λ1(aΩ), we must have that Aμ,λ(ϕ1) > 0, which implies that Aμ,λ(ϕ1), B(ϕ1) > 0 for all μ > μ0(λ). Hence t(ϕ1)ϕ1Nμ,λ, where

t(ϕ1)=RN(|ϕ1|2dxλaϕ12)dxBϕ11/2p2=(λ1a¯Ωλ)RNaϕ12dxBϕ11/2p2>0.

Thus,

Jμ,λ(t(ϕ1)ϕ1)=p22p(λ1a¯Ωλ)RNaϕ12dxp/p1Bϕ11/p10 as λλ1a¯Ω.

Since 0 < infuNμ,λ Jμ,λ(u) ≤ Jμ,λ(t(ϕ1)ϕ1), it follows that limλλ1ainfuNμ,λJμ,λ(u) = 0. This completes the proof. □

Next, we are ready to prove Theorem 1.2:

  1. Since λnλ1a¯Ω and μ͠0(λn) → ∞ as n → ∞, we have μn → ∞ as n → ∞. Firstly, we show that {un} is bounded. Suppose on the contrary. Then we may assume without loss of generality that ∥unμn → ∞ as n → ∞. Let vn = ununμn. Since ∥vnμn = 1, by Lemma 2.4, there exist subsequence {vn} and v0H01(Ω) such that vnv0 in Lr(ℝN) for 2 ≤ r < 2 and B(vn) → B(v0). Hence

    limnRNavn2dx=RNav02dx.

    By Theorem 3.3,

    Jμn,λn(un)=p22punμn2λnRNaun2dx=p22pBun0 as n,

    dividing by unμn2 it is easy to see that

    limnvnμn2λnRNavn2dx=0

    and

    limnunμn2p2Bvn=0.

    Thus,

    limnλnRNavn2dx=λ1a¯ΩRNav02dx=1

    and

    limnBvn=Bv0=0.

    Now, we show that

    limnRN|vn|2dx=RN|v0|2dx.

    If not, then we may assume that

    0RN(|v0|2λ1a¯Ωav02)dx<lim infnvnμn2λnRNavn2dx=0

    which is impossible. Thus, we must have

    Ω(|v0|2λ1a¯Ωav02)dx=limnvnμn2λnRNavn2dx=0,

    and so v0 = 1 for some k. Since B(v0) = |k|2pB(ϕ1) = 0 and B(ϕ1) > 0, it follows that k = 0. But, as RNav02dx0, this is impossible. Hence {un} is bounded. By Lemma 2.4, we may assume that there exists u0H01(Ω) such that

    limnRNaun2dx=RNau02dx and limnBun=Bu0.

    Moreover, by Theorem 3.3,

    Jμn,λn(un)=p12pAμn,λnun=p22pBun0 as n,

    which indicates that

    limnBun=Bu0=0.

    Since

    0RN(|u0|2λ1a¯Ωau02)dxlim infnAμn,λnun=0,

    and so u0 = 1 for some k. Since B(u0) = |k|2pB(ϕ1) = 0 and B(ϕ1) > 0, it follows that k = 0 and u0 ≡ 0. Therefore, un → 0 in X.

  2. Here we follow the argument in [9] (or see [45]) to investigate the concentration for positive solutions of Eq. (Pμ,λ). For any sequence μn → ∞, let un := uμn,λ be the positive solutions of Eq. (Pμn,λ) obtained in Theorem 1.1. By Lemma 2.4 there exists a positive constants c0 and C0 are independent of μn such that ∥unμnc0 and Jμn,λ(un) ≥ C0. Therefore, by Lemma 2.4, we may assume that there exists u0H01(Ω) such that unu0 in X and unu0 in Lr(ℝN) for all 2 ≤ r < 2. Now for any φC0(Ω), because Jμn,λun,φ = 0, it is easy to check that

    Ωu0φdx=λΩa¯Ωu0φdx+Ωfu0p2u0φdx,

    that is, u0 is a weak solution of Eq. (P) by the density of C0(Ω) in H01(Ω). Now, we show that unu0 in X. Because Jμn,λun,un=Jμn,λun,u0 = 0, we have

    unμn2=λRNaun2dx+RNfunpdx(3.11)

    and

    un,u0μn=λRNaunu0dx+RNfunp2unu0dx.(3.12)

    By (3.11), (3.12) and unu0 in Lr(ℝN) for all 2 ≤ r < 2, we have

    limnunμn2=limnun,u0μn=limnun,u0=u02.

    On the other hand, the weakly lower semi-continuity of norm yields

    u02lim infnun2limnunμn2,

    and thus, unu0 in X. Moreover, by Jμn,λ(un) ≥ C0 > 0, one has u0 ≠ 0, which completes the proof.

4 The Proof of Theorems 1.3, 1.4 (λ > λ1(aΩ))

If λ > λ1(aΩ), then

Aμ,λϕ1=RN(|ϕ1|2λaϕ12)dx=(λ1a¯Ωλ)RNaϕ12dx<0 for all μ>0.

Hence, if B(ϕ1) < 0, then by Lemma 2.1, Nμ,λ+ ≠ ∅. Thus, as well shall see, Nμ,λ may consist of two distinct components in this case which makes it possible to prove the existence of at least two positive solutions by showing that Jμ,λ has an appropriate minimizer on each component.

If λ is just greater than λ1(aΩ), then roughly speaking uμ2λRNau2dx if and only if u is almost a multiple of ϕ1 for μ enough large. Thus, if B(ϕ1) < 0, it should follow that does not there exists uX \ {0} such that Aμ,λ(ϕ1) ≤ 0 and B(ϕ1) ≥ 0. This is made precise in the following lemma and we show subsequently that Nμ,λ0=(orNμ,λ=Nμ,λ+Nμ,λ) is an important condition for establishing the existence of minimizers.

Let

Aμ,λ=uX0:Aμ,λu0

and

Bμ,λ=uX0:Bu0.

Then we have

Lemma 4.1

Suppose that B(ϕ1) < 0. Then there exist δ0 > 0 and μ̂0μ0 such that 𝓐μ,λ ∩ 𝓑μ,λ = ∅ for all λ1(aΩ) < λ < λ1(aΩ) + δ0 and μ > μ̂0. In particular, Nμ,λ0 = ∅ for all λ1(aΩ) < λ < λ1(aΩ) + δ0 and μ > μ̂0.

Proof

Suppose that the result is false. Then there exist sequences {μn}, {λn} and {wn} ⊂ X \ {0} with λnλ1+(aΩ) and μn → ∞ such that

Aμn,λnwn=wnμn2λnRNawn2dx0

and

Bwn=RNIαwnpwnpdx+RNf|wn|2pdx0.

Let un = wnwnμn. Since ∥un∥ ≤ ∥unμn = 1, by Lemma 2.4, we may assume that there exists u0H01(Ω) such that unu0 a.e. in ℝN, unu0 in Lr(ℝN) for all 2 ≤ r < 2 and B(un) → B(u0). Then

limnλnRNaun2dx=λ1+a¯ΩRNau02dx1.(4.1)

Now, we show that limn→∞Ω|∇ un|2dx = ∫Ω|∇ u0|2dx. Suppose on the contrary. Then by (4.1),

Ω|u0|2λ1a¯Ωa¯Ωu02dx=RN|u0|2λ1a¯Ωau02dx<lim infnunμn2λnRNaun2dx0,

which is impossible. Hence limn→∞Ω|∇ un|2dx = ∫Ω|∇ u0|2dx. It follows that

(I)Ω(|u0|2λ1a¯Ωa¯Ωu02)dx0,(II)Bu00.

But (I) implies that u0 = 1 for some k and then (II) implies that k = 0 which is impossible as λ1+(aΩ) ∫N au02dx ≥ 1. Thus, there exists δ0 > 0 and μ̂0μ0 such that 𝓐μ,λ ∩ 𝓑μ,λ = ∅ for all λ1(aΩ) < λ < λ1(aΩ) + δ0 and μ > μ̂0. Moreover, if Nμ,λ0 ≠ ∅, then there exists u0Nμ,λ0 such that u0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Therefore, Nμ,λ0 = ∅ for all λ1(aΩ) < λ < λ1(aΩ) + δ0 and μ > μ̂0. This completes the proof. □

When Nμ,λ0 = ∅, any non-zero minimizer for Jμ,λ on Nμ,λ+ (or on Nμ,λ) is also a local minimizer on Nμ,λ and so will be a critical point for Jμ,λ on Nμ,λ and a solution of Eq. (Pμ,λ). We next show that, if Nμ,λ0 = ∅, it is possible to obtain more information about the nature of the Nehari manifold. Since B(ϕ1) < 0, we can obtain that Nμ,λ+ ≠ ∅ for all μ > 0. Furthermore, we have the following results.

Lemma 4.2

Suppose that B(ϕ1) < 0. Then for any λ1(aΩ) < λ < λ1(aΩ) + δ0 and for μ enough large, we have the following results.

  1. Nμ,λ+ is uniform bounded.

  2. There exist two negative numbers κ1 and κ2 such that

    κ1infuNμ,λ+Jμ,λ(u)<κ2.

Proof

  1. Suppose on the contrary. Then there exist sequences μnR+Nand{un}Nμn,λ+ such that μn → ∞ and ∥unμn → ∞ as n → ∞. Clearly,

    Aμn,λun=Bun<0.(4.2)

    Let vn = ununμn. Then by Lemma 2.4, we may assume that there exists v0H01(Ω) such that

    vnv0 in X;vnv0 in LrRN for all 2r<2,

    and

    limnBvn=Bv0.(4.3)

    Thus,

    limnRNavn2dx=RNav02dx.(4.4)

    Moreover, by Fatou’s Lemma,

    RN|v0|2dxlim infnRN|vn|2dx.(4.5)

    Dividing (4.2) by unμn2 gives

    Aμn,λvn=unμnp2Bvn<0.(4.6)

    Since

    limnAμn,λvn=1λlimnRNavn2dx=1λRNav02dx

    and ∥unμn → ∞, it obtain that B(v0) = 0 and ∫N av02dx > 0 from the conclusions (4.3) and (4.6). Thus, v0 ∈ 𝓑μ,λ for all μ > 0. Moreover, by v0H01(Ω), (4.5) and (4.4), for every μ > 0,

    v0μ2λRNav02dx=RN|v0|2λav02dx<lim infnAμn,λvn0,

    which indicates that v0 ∈ 𝓐μ,λ. We now show that vnv0 in Xμ. Suppose on the contrary. Then

    λv0μ2RNav02dx=RN|v0|2λav02dx<limnAμn,λvn0,

    since ∫N gv02dx = 0. Hence v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Since vnv0 in Xμ, then ∥v0μ = 1. Hence v0 ∈ 𝓑μ,λ. Moreover,

    v0μ2λRNav02dx=limnAμn,λvn0

    and so v0 ∈ 𝓐μ,λ. Thus, v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Hence Nμ,λ+ is uniform bounded for μ > 0 sufficiently large.

  2. By part (i), there exists C0 > 0 such that ∥uμC0 for all uNμ,λ+. Hence, making use of (2.1), for uNμ,λ+ we have

    Jμ,λ(u)=p12pBup12pRNIαupupdx+fRNu2pdxp12pC1uμ2pp12pSpC1C0p=κ1.(4.7)

    Moreover, by B(ϕ1) < 0 and ∫Ω|∇ ϕ1|2dxλΩ aϕ12dx < 0, which indicates that the function hϕ1(t) = Jμ,λ(1) have t0+ > 0 and κ2 < 0 are independent of μ such that t0+φNμ,λ+ and

    inf0<t<hϕ1t=hϕ1t0+=κ2<0.

    This implies that

    infuNμ,λ+Jμ,λuκ2<0 for all μ>maxμ¯1,μ¯2.(4.8)

    This completes the proof. □

Theorem 4.3

Suppose that B(ϕ1) < 0. Then for any λ1(aΩ) < λ < λ1(aΩ) + δ0 and for μ enough large, there exists a minimizer of Jμ,λ(u) on Nμ,λ+.

Proof

By Lemmas 4.1, 4.2 and the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ Nμ,λ+ such that

limnJμ,λ(un)=infuNμ,λ+Jμ,λ(u)κ2 and Jμ,λ(un)=o1

and there exists C0 > 0 such that ∥unμC0. Thus, there exist a subsequence {un} and u0Xμ such that Jμ,λ (u0) = 0 and

unu0 in Xμ,unu0 a.e. in RN,unu0 in LlocrRN for 2r<2.

Then by condition (V4),

limnRNaun2dx=RNau02dx.(4.9)

Moreover, follows from Brezis–Lieb lemma [8] and Lemma 2.3, obtain that

Bunu0=BunBu0+o(1).(4.10)

Now we prove that unu0 in Xμ. Let vn = unu0. Then vn ⇀ 0 in Xμ. By the Sobolev and Gagliardo–Nirenberg inequalities, for any μ > μ0 we have that

RNvn2dx1μcgcμgvn2dx+g<cvn2dx1μcRNμgvn2dx+o1

and

RNvn2pdxC01μcvnμ222p22RNvn2dx2p122+o1C01μc22p22vnμ2p+o1

or

RNvn2pdxΠμvnμ2p+o(1),(4.11)

where Πμ=C01μc22p22. Thus, using (4.9) and (4.10) gives

Jμ,λvn=Jμ,λunJμ,λu0+o1 and Jμ,λ(vn),vn=o(1).(4.12)

Consequently, by (4.7), (4.9), (4.12) and Lemma 4.2 (ii), one has

κ2+κ1infuNμ,λ+Jμ,λ(u)Jμ,λu0Jμ,λvn12pJμ,λ(vn),vn+o1p12pvnμ2+o(1),

which shows that there exists a constant C1 > 0 such that

vnμC1+o1 for μ>0 sufficiently large.(4.13)

Since 1 < N+αN<p<2α, it follows from (4.9), (4.11) and (4.13) that

o1=Jλ(vn),vnvnμ21C0Πμvnμ2p2+o(1)vnμ21fΠμC12p2+o1.(4.14)

Notice that Πμ → 0 as μ → ∞. Then by (4.14), there holds vn → 0 in Xμ for μ > 0 sufficiently large. Hence unu0 in Xμ and so

Jμ,λ(u0)=limnJμ,λ(un)=infuNμ,λ+Jμ,λ(u)κ0<0,

which implies that u0 is a minimizer on Nμ,λ+. □

We now turn our attention to Nμ,λ.

Lemma 4.4

Suppose that B(ϕ1) < 0. Then for any λ1(aΩ) < λ < λ1(aΩ) + δ0 and for μ enough large, we have the following results

  1. there exists c0 > 0 such thatuμc0 for all uNμ,λ;

  2. every minimizing sequence for Jμ,λ(u) on Nμ,λ is bounded;

  3. infuNμ,λ Jμ,λ(u) > 0.

Proof

  1. Suppose on the contrary. Then there exist {μn} ⊂ ℝ+ and {un} ⊂ Nμn,λ such that μn → ∞ and ∥unμn → 0. Hence, by (2.1),

    0<Aμn,λun=Bun0 as n.

    Let vn = ununμn. Then, by Lemma 2.4, there exist subsequence {vn} and v0H01(Ω) such that

    vnv0 in X;vnv0 in LrRN for all 2r<2.

    Thus,

    limnRNavn2dx=RNav02dx(4.15)

    and

    Aμn,λvn=unμn2p2Bvn0 as n.(4.16)

    Moreover, by (4.15), (4.16), v0H01(Ω) and Fatou’s Lemma, we can obtain that

    0=limnAμn,λvn=1λlimnRNavn2dx=1λRNav02dx,

    and for every μ > 0

    v0μ2RNλav02dx=RN|v0|2λav02dxlim infnvnμn2RNλavn2dx=0,

    this implies that v0 ≠ 0 and v0 ∈ 𝓐μ,λ for all μ > 0. Since B(vn) > 0 and B(vn) → B(v0), it follows that v0v0μ ∈ 𝓑μ,λ for all μ > 0. Hence, v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ for all μ > 0, which a contradiction.

  2. Suppose on the contrary. Then there exist sequences {μn} ⊂ ℝ+ with μn → ∞ such that Nμn,λ is unbounded for all n, that is for every n there exists a minimizing sequence {un,m} ⊂ Nμn,λ such that ∥un,mμn → ∞ as m → ∞. Moreover,

    Aμn,λun,m=Bun,mp12pinfuNμn,λJμn,λ(u) as m,(4.17)

    where infuNμn,λ Jμn,λ(u) ≥ 0 for all n. Let wn = un,n. Then wnNμn,λ and ∥wnμn → ∞ as n → ∞. Let vn = wnwnμn. Then by Lemma 2.4, we may assume that there exist subsequence {vn} and v0H01(Ω) such that vnv0 in X, vnv0 in Lr(ℝN) for all 2 ≤ r < 2 and B(vn) → B(v0). Then by condition (V4)

    limnRNavn2dx=RNav02dx.(4.18)

    Dividing (4.17) by wnμn2 and m = n gives

    Aμn,λvn=wnμnp2Bvn0.(4.19)

    Since ∥wnμn → + ∞, it follows that B(vn) → 0 and so B(v0) = 0. We now show that vnv0 in X. Suppose otherwise, then by (4.18) and (4.19),

    RN|v0|2λav02dx=v02λRNav02dx<lim infnvnμn2λRNavn2dx=0.

    Thus, v0 ≠ 0 and for every μ > 0, there holds v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ, which is impossible. Hence vnv0 in X. It follows that ∥v0μ = 1, ∫NVv02dx = 0 and

    v0μ2λRNav02dx=Bv0=0.

    Thus, for every μ > 0, there holds v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible as 𝓐μ,λ ∩ 𝓑μ,λ = ∅. Hence, every minimizing sequence for Jμ,λ(u) on Nμ,λ is bounded for μ sufficiently large.

  3. Assume that infuNμ,λ Jμ,λ(u) = 0. Then by the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ Nμ,λ such that

    limnJμ,λ(un)=infuNμ,λJμ,λ(u) and Jμ,λ(un)=o1.

    By part (ii), {un} is bounded and so there exist a subsequence {un} and u0Xμ such that Jμ,λ(u0) = 0 and

    unu0 in Xμ,unu0 a.e. in RN,unu0 in LlocrRN for 2r<2.

    Then by condition (V4)

    limnRNavn2dx=RNav02dx.(4.20)

    Moreover, follows from Brezis–Lieb lemma [8] and Lemma 2.3, obtain that

    Bunu0=BunBu0+o(1).(4.21)

    Now we prove that unu0 in Xμ. Let vn = unu0. Then vn ⇀ 0 in Xμ. By the Sobolev and Gagliardo–Nirenberg inequalities, for μ enough large we have that

    RNvn2dx1μcgcμgvn2dx+g<cvn2dx1μcRNμgvn2dx+o1

    and

    RNvn2pdxC01μcvnμ222p22RNvn2dx2p122+o1C01μc22p22vnμ2p+o1.

    or

    RNvn2pdxΠμvnμ2p+o(1),(4.22)

    where Πμ=C01μb22p22SNp1. Thus, using (4.21) and unu0 in Xμ gives

    Jμ,λvn=Jμ,λunJμ,λu0+o1 and Jμ,λ(vn),vn=o(1).(4.23)

    Consequently, by (4.20), (4.21) and (4.23), one has

    infuNμ,λJμ,λ(u)Jμ,λu0Jμ,λvn12pJμ,λ(vn),vn+o1p12pvnμ2+o(1).(4.24)

    Suppose that infuNμ,λ Jμ,λ(u) = 0.

    (iiiA) If u0Nμ,λ, then by (4.24) and u0 = 0, vnμ2 → 0, this shows that unu0 in Xμ, and so

    Jμ,λ(u0)=limnJμ,λ(un)=infuNμ,λJμ,λ(u)=0.

    It then follows exactly as in the proof in part (i) that u0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible as 𝓐μ,λ ∩ 𝓑μ,λ = ∅.

    (iiiB) If u0Nμ,λ+, then by (4.7) and (4.24), there exists C0 > 0 such that

    vnμC0+o1 for μ enough large.(4.25)

    Since 1 < N+αN<p<2α, it follows from (4.18), (4.22) and (4.25) that

    o1=Jμ,λ(vn),vnvnμ21fΠμvnμ2p2+o(1)vnμ21fΠμC02p2+o1.(4.26)

    Notice that Πμ → 0 as μ → ∞. Then by (4.26), for μ enough large, there holds vn → 0 in Xμ. Hence unu0 in Xμ, and so u0Nμ,λ this is a contradiction. Thus, infuNμ,λ Jμ,λ(u) > 0 for μ enough large. This completes the proof. □

Theorem 4.5

Suppose that B(ϕ1) < 0. Then for any λ1(aΩ) < λ < λ1(aΩ) + δ0 and for μ enough large, there exists a minimizer of Jμ,λ(u) on Nμ,λ.

Proof

By Lemmas 4.1, 4.4 (iii) and the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ Nμ,λ such that

limnJμ,λ(un)=infuNμ,λJμ,λ(u) and Jμ,λ(un)=o1.

Similar the argument in (3.3), there exists D0 > 0 independent of μ such that infuNμ,λ Jμ,λ(u) < D0 for all μμ0. Moreover, by Lemma 4.4 (ii), there exists C0 > 0 such that ∥unμC0. Thus, there exist a subsequence {un} and u0Xμ such that Jμ,λ(u0) = 0 and

unu0 in Xμ,unu0 a.e. in RN,unu0 in LlocrRN for 2r<2.

Then by condition (V4),

limnRNavn2dx=RNav02dx,(4.27)

and follows from Brezis–Lieb lemma [8] and Lemma 2.3 obtain that

Bunu0=BunBu0+o(1).(4.28)

Now we prove that unu0 in Xμ. Let vn = unu0. Then vn ⇀ 0 in Xμ. By the Sobolev and Gagliardo–Nirenberg inequalities, for μ enough large we have that

RNvn2dx1μcgcμgvn2dx+g<cvn2dx1μcRNμgvn2dx+o1

and

RNvn2pdxC01μcvnμ222p22RNvn2dx2p122+o1C01μc22p22vnμ2p+o1.

or

RNvn2pdxΠμvnμ2p+o(1),(4.29)

where Πμ=C01μc22p22. Thus, using (4.27) and (4.28) gives

Jμ,λvn=Jμ,λunJμ,λu0+o1 and Jμ,λvn=o(1).(4.30)

Consequently, by (4.27), (4.30) and Lemma 4.2 (ii), one has

D0+κ1infuNμ,λJμ,λ(u)Jμ,λu0Jμ,λvn1pJμ,λ(vn),vn+o1p22pvnμ2+o(1),(4.31)

which shows that there exists a constant C1 > 0 such that for μ enough large,

vnμC1+o1.(4.32)

Since 1<N+αN<p<2α, it follows from (4.29) (4.31) and (4.32) that

o1=Jλ(vn),vnvnμ21fΠμvnμ2p2+o(1)vnμ21fΠμC12p2+o1.(4.33)

Notice that Πμ → 0 as μ → ∞. Then by (4.33), for μ enough large, there holds vn → 0 in Xμ. Hence unu0 in Xμ and so

Jμ,λ(u0)=limnJμ,λ(un)=infuNμ,λJμ,λ(u),

which implies that u0 is a minimizer on Nμ,λ.□

We are now ready to prove Theorem 1.3: By Theorem 4.5 and 4.3, there exist δ0 such that when λ 1(aΩ) < λ < λ1(aΩ) + δ0 and for μ enough large, Jμ,λ has minimizers in each of Nμ,λ1 and Nμ,λ2, that is there exist uμ,λ1Nμ,λ+ and uμ,λ2Nμ,λ such that

Jμ,λ(uμ,λ1)=infuNμ,λ+Jμ,λ(u)<κ2<0<infuNμ,λJμ,λ(u)=Jμ,λ(uμ,λ2).

Since Jμ,λ(uμ,λj)=Jμ,λ(|uμ,λj|) for j = 1, 2, we may assume that these minimizers are positive. Moreover, by Lemma 4.1, Nμ,λ=Nμ,λ+Nμ,λ. It follows that the minimizers are local minimizers in Nμ,λ which do not lie in Nμ,λ0, and so by Lemma 2.2, uμ,λ1 and uμ,λ2 are positive solutions of Eq. (Pμ,λ). This completes the proof.

Next, we are ready to prove Theorem 1.4: (i) Since Nμn,λn+ is uniformly bounded, then {un} is bounded, from Lemma 2.4, we may assume that there exists u0H01(Ω) such that unu0 in X, unu0 in Lr(ℝN) for all 2 ≤ r < 2 and B(un) → B(u0). We also have

limnRNaun2dx=RNau02dx

and

un2λnRNaun2dxAμn,λnun=Bun<0 for n sufficiently large.

We now show that unu0 in X. Suppose on the contrary. Then

RN(|u0|2λ1a¯Ωau02)dx=RN(|u0|2+Vu02λ1a¯Ωau02)dx<lim infnun2λnRNaun2dx0,

which is impossible. Thus, unu0 in X and so

Ω(|u0|2λ1a¯Ωau02)dxRN(|u0|2+Vu02λ1a¯Ωau02)dx=Bu00,

this implies that Ω(|u0|2λ1a¯Ωa¯Ωu02)dx=0 and we must have u0 = 1 for some k. But, as B(ϕ1) < 0, it follows that k = 0. Therefore, un → 0 in X. Next, let vn=ununμn. Then by Lemma 2.4, we may assume that there exists v0H01(Ω) \ {0} such that vnv0 in X, vnv0 in Lr(ℝN) for all 2 ≤ r < 2 and B(vn) → B(v0). Thus,

limnRNavn2dx=RNav02dx.(4.34)

Clearly,

vn2λnRNavn2dxAμn,λnvn=unμn2p2Bvn<0(4.35)

for n sufficiently large. We now show that vnv0 in X. Suppose on the contrary. Then by (4.34) and (4.35),

Ω(|v0|2λ1a¯Ωa¯Ωv02)dx=RN(|u0|2+Vv02λ1a¯Ωa¯Ωv02)dx<lim infnvn2λnRNavn2dx0,

and so

Ω(|v0|2λ1a¯Ωa¯Ωv02)dx<0,

which gives a contradiction. Hence vnv0 in X, which indicates that ∫Ω∣∇ v02dx = 1 and

Ω(|v0|2λ1a¯Ωa¯Ωv02)dx=0.

Therefore, v0 = ϕ1.

(ii) For any sequence μn → ∞, let unj:=uμnj (j = 1, 2) be the solutions obtained in Theorem 1.3 with uμn1Nμn,λ+ and uμn2Nμn,λ. Similar to the argument of proofs in Lemma 4.4 (ii) and Lemma 4.2 (i) there exists a positive constant c0 is independent of μn such that

unjμnc0.(4.36)

Therefore, by Lemma 2.4, we may assume that there exist u0jH01Ω such that unju0j in X and BunjBu0j. Now for any φC0 (Ω), because Jμn,λunj,φ=0, it is easy to check that

Ωu0jφdxλΩa¯Ωu0jφdx+ΩIαu0jpu0jp2u0jφdx=Ωfu0j2p2u0jφdx,

that is, u0j are weak solutions of Eq. (P) by the density of C0(Ω) in H01(Ω). Now, we show that unju0j in X for j = 1, 2. Because Jμn,λunj,unj=Jμn,λunj,u0j=0, we have

unjμn2λRNaunj2dx+ΩIαunjpunjpdx=RNfunjpdx(4.37)

and

unj,u0jμnλRNaunju0jdx+ΩIαunjpunjp2unju0jdx=RNfunjp2unju0jdx.(4.38)

By (4.36)(4.38) and unju0j in Lr(ℝN) for all 2 ≤ r < 2, we have

limnunjμn2=limnunj,u0jμn=limnunj,u0j=u0j2.

On the other hand, the weakly lower semi-continuity of norm yields

u0j2lim infnunj2limnunjμn2,

and thus, unju0j in X for j = 1, 2. By Lemma 4.4 (i) and the fact that un2μn>c0>0, which implies that u020. Moreover, by (4.8), there exists κ0 < 0

Jμn,λun1=infuNμn,λ+Jμn,λuκ0 for all n.

Thus,

Jμ,λ|H01Ωu01κ0<0,

which implies that u010. To complete the proof, it remains to show that u01andu02 are distinct. That Jμ,λ|H01Ωu01κ0<0andJμ,λ|H01Ωu02>0 implies that u01u02. This completes the proof.□

Acknowledgement

This research was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 106-2115-M-390-002-MY2) and the National Center for Theoretical Sciences, Taiwan.

References

  • [1]

    H. Amann and J. Lopez-Gomez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146, (1998), 336–374.CrossrefGoogle Scholar

  • [2]

    A. Azzollini, Concentration and compactness in nonlinear Schrödinger–Poisson system with a general nonlinearity, J. Differential Equations 249, (2010), 1746–1763. CrossrefGoogle Scholar

  • [3]

    C. O. Alves and A. B. Nobrega, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Diff. Equ. (2016), 55:48. CrossrefGoogle Scholar

  • [4]

    A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commum. Contemp. Math. 10 (2008), 391–404. CrossrefGoogle Scholar

  • [5]

    P.A. Binding, P. Drabek, Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations 5, (1997), 1–11. Google Scholar

  • [6]

    P.A. Binding, P. Drabek, Y.X. Huang, Existence of multiple solutions of critical quasilinear elliptic Neumann problems, Nonlinear Anal. 42, (2000), 613–629.CrossrefGoogle Scholar

  • [7]

    V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11, (1998), no. 2, 283–293. CrossrefGoogle Scholar

  • [8]

    H. Brezis and E. Lieb, A relation between point convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88, (1983), 486–490. CrossrefGoogle Scholar

  • [9]

    T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3, (2001), 549–569. CrossrefGoogle Scholar

  • [10]

    T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discr. Cont. Dyna. Systems A 33, (2013), 7–26. Google Scholar

  • [11]

    T. Bartsch and Z. Q. Wang, Existence and multiplicity results for superlinear elliptic problems on ℝN, Comm. Partial Differential Equations 20, (1995), 1725–1741. CrossrefGoogle Scholar

  • [12]

    K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193, (2003), 481–499. CrossrefGoogle Scholar

  • [13]

    J. Chabrowski and D.G. Costa, On a class of Schrödinger–Type equations with indefinite weight functions, Comm. Partial Differential Equations 33, (2008), 1368–1394. CrossrefGoogle Scholar

  • [14]

    J. Chen, Multiple positive solutions of a class of non autonomous Schrödinger–Poisson systems, Nonlinear Analysis: Real World Appl. 21, (2015), 13–26. CrossrefGoogle Scholar

  • [15]

    D.G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in ℝN, Calc. Var. Partial Diff. Equ. 13, (2001), 159–189.CrossrefGoogle Scholar

  • [16]

    G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differential Equations 248, (2010), 521–543. CrossrefGoogle Scholar

  • [17]

    P. Drábek and S.I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh A 127, (1997), 703–726. CrossrefGoogle Scholar

  • [18]

    Y. Deng and W. Shuai, Sign-changing multi-bump solutions for Kirchhoff-type equations in ℝN, Discr. Cont. Dyna. Systems A 38, (2018), 3139–3168. CrossrefGoogle Scholar

  • [19]

    I. Ekeland, Convexity methods in Hamiltonian mechanics, Springer, 1990. Google Scholar

  • [20]

    R.L. Frank and E.H. Lieb, Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality, Calc. Var. Partial Differ. Equ. 39, (2010), 85–99. CrossrefWeb of ScienceGoogle Scholar

  • [21]

    L. Huang, E. M. Rocha, J. Chen, Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity, J. Differential Equations 255, (2013), 2463–2483. CrossrefGoogle Scholar

  • [22]

    L. Huang, E.M. Rocha, J. Chen, On the Schrödinger–Poisson system with a general indefinite nonlinearity, Nonlinear Analysis: Real World Appl. 28, (2016), 1–19. CrossrefGoogle Scholar

  • [23]

    I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger–Poisson problems with potentials, Adv. Nonlinear Stud. 8, (2008), 573–595. Google Scholar

  • [24]

    Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations 251, (2011), 582–608. CrossrefGoogle Scholar

  • [25]

    E.H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118, (1983), 349–374. CrossrefGoogle Scholar

  • [26]

    E.H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Math. 14, American Mathematical Society, Providence, 2001. Google Scholar

  • [27]

    P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann. Inst. H. Poincare Anal. Non Lineaire 1, (1984), 109–145. CrossrefGoogle Scholar

  • [28]

    V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265, (2013), 153–184. Web of ScienceCrossrefGoogle Scholar

  • [29]

    Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95, (1960), 101–123. CrossrefGoogle Scholar

  • [30]

    D. Ruiz, The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237, (2006), 655–674. CrossrefGoogle Scholar

  • [31]

    D. Ruiz, On the Schrödinger–Poisson–Slater system: behavior of minimizer, radial and nonradial cases, Arch. Ration. Mech. Anal. 198, (2010), 349–368. CrossrefGoogle Scholar

  • [32]

    Z. Shen and Z. Han, Multiple solutions for a class of Schrödinger–Poisson system with indefinite nonlinearity, J. Math.Anal.Appl. 426, (2015), 839–854. CrossrefGoogle Scholar

  • [33]

    J. Sun and T.F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations 256, (2014), 1771–1792. CrossrefWeb of ScienceGoogle Scholar

  • [34]

    J. Sun and T.F. Wu, On the nonlinear Schrödinger–Poisson systems with sign–changing potential, Z. Angew. Math. Phys. 66, (2015), 1649–1669. CrossrefGoogle Scholar

  • [35]

    J. Sun, T.F. Wu, Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differential Equations 260, (2016), 586–627. CrossrefGoogle Scholar

  • [36]

    J. Sun, T.F. Wu, Z. Feng, On the non-autonomous Schrödinger–Poisson problems in ℝ3, Discrete Contin. Dyn. Syst. A 38, (2018), 1889–1933. Google Scholar

  • [37]

    J. Sun, T.F. Wu, Y. Wu, Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well, Z. Angew. Math. Phys. 68, (2017), 1–22. Google Scholar

  • [38]

    C. Stuart and H. Zhou, Global branch of solutions for nonlinear Schrödinger equations with deepening potential well, Proc. Lond. Math. Soc. 92, (2006), 655–681. CrossrefGoogle Scholar

  • [39]

    N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20, (1967), 721-747. CrossrefGoogle Scholar

  • [40]

    G. Vaira, Ground states for Schrödinger–Poisson type systems. Ric. Mat. 60, (2011), 263–297. CrossrefGoogle Scholar

  • [41]

    G. Vaira, Existence of bound states for Schrödinger–Newton type systems. Adv. Nonlinear Stud. 13, (2013), 495–516. Google Scholar

  • [42]

    M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar

  • [43]

    Z. Wang and H. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Europ. Math. Soc. 11, (2009), 545–573. Google Scholar

  • [44]

    T. F. Wu, The Nehari manifold for indefinite nonlinear Schrödinger equations in ℝN, preprint. Google Scholar

  • [45]

    L. Zhao, H. Liu, F. Zhao, Existence and concentration of solutions for the Schrödinger–Poisson equations with steep well potential, J. Differential Equations 255, (2013), 1–23. CrossrefGoogle Scholar

  • [46]

    X. Zhang and S. Ma, Multi–bump solutions of Schrödinger–Poisson equations with steep potential well, Z. Angew. Math. Phys. 66, (2015), 1615–1631. CrossrefGoogle Scholar

About the article

Received: 2018-11-10

Accepted: 2019-02-14

Published Online: 2019-07-20

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 665–689, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0020.

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© 2020 Tsung-fang Wu, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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