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Volume 9, Issue 1

# 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.

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

$−Δu+Vμ,λxu+Iα∗upup−2u=f(x)u2p−2u in RN,u∈H1RN,$(Pμ,λ)

where $\begin{array}{}N\phantom{\rule{thinmathspace}{0ex}}\ge \phantom{\rule{thinmathspace}{0ex}}3,\frac{N+\alpha }{N}\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}\frac{N}{N-2}\end{array}$ and Iα is the Riesz potential of order 0 < α < min {N, 2} $\begin{array}{}\left({2}^{\ast }=\frac{2N}{N-2}\right)\end{array}$ on the Euclidean space ℝN, defined for each point x ∈ ℝN∖{0} by

$Iαx=Γ(N−α2)Γ(α2)πN/22αxN−α$

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

$−Δu(x)=λa¯Ω(x)u(x)forx∈Ω;u(x)=0forx∈∂Ω,$(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:

$−Δu+Vxu+σ(I∗u2)u=f(x)u2p−2u in R3,$(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:

$−△u+V(x)u+σϕu=f(x)u2p−2u,in R3,−△ϕ=u2,in R3.$(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 = $\begin{array}{}\frac{3}{2}\end{array}$ 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:

$Nr={u∈Hr1(R3)∖{0}:Q(u)=0},$

where $\begin{array}{}{H}_{r}^{1}\end{array}$(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

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

$−△u+1−λhxu+lxϕu=f(x)u2p−2u,in R3,−△ϕ=lxu2,in R3,$(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)={u∈N:I(u)

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

$N(c)=N1∪N2,$

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 $\begin{array}{}\frac{1+\sqrt{73}}{6}\end{array}$ < 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 $\begin{array}{}\underset{\lambda \to {\lambda }_{1}^{-}\left({\overline{a}}_{\mathit{\Omega }}\right)}{lim}\end{array}$ 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 λ$\begin{array}{}{\lambda }_{1}^{-}\end{array}$ (aΩ) or μ → ∞. Let

$Bu:=−∫RNIα∗upupdx+∫RNfu2pdx.$

Then we have the following result.

#### Theorem 1.2

1. Suppose that B(ϕ1) > 0. Let λn$\begin{array}{}{\lambda }_{1}^{-}\end{array}$ (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,λ$\begin{array}{}{u}_{\lambda }^{\mathrm{\infty }}\end{array}$ in X as μ → ∞, where $\begin{array}{}{u}_{\lambda }^{\mathrm{\infty }}\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ is a positive solution of

$−Δu−λa¯Ωxu+Iα∗upup−2u=fxu2p−2uinΩ,u=0,on∂Ω.$(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 $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\end{array}$ and $\begin{array}{}{u}_{\mu ,\lambda }^{\left(2\right)}\end{array}$ with $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\end{array}$ is negative energy and $\begin{array}{}{u}_{\mu ,\lambda }^{\left(2\right)}\end{array}$ is positive energy. Furthermore, $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\end{array}$ is the least energy positive solution of Eq. (Pμ,λ).

Finally, we investigate the nature of least energy positive solution $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\end{array}$ as λ$\begin{array}{}{\lambda }_{1}^{+}\end{array}$ (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 $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\end{array}$ 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 λn$\begin{array}{}{\lambda }_{1}^{+}\end{array}$ (aΩ) and μn → ∞ be as in Theorem 1.3 and let $\begin{array}{}{u}_{n}^{\left(1\right)}:={u}_{{\mu }_{n},{\lambda }_{n}}^{\left(1\right)}\end{array}$ be the least energy positive solutions of Eq. (Pμn,λn) obtained by Theorem 1.3. Then

$un→0;un∥un∥μn→ϕ1inXasn→∞.$

2. For λ1 (aΩ) < λ < λ1 (aΩ) + δ0. Let $\begin{array}{}{u}_{\mu ,\lambda }^{\left(j\right)}\end{array}$ (j = 1, 2) be the positive solutions obtained in Theorem 1.3. Then $\begin{array}{}{u}_{\mu ,\lambda }^{\left(j\right)}\to {u}_{\lambda }^{\left(j\right),\mathrm{\infty }}\end{array}$ in X as μ → ∞, where $\begin{array}{}{u}_{\lambda }^{\left(j\right),\mathrm{\infty }}\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ are positive solutions of Eq. (P).

#### Remark 1.2

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

$−Δu+Vμ,λxu=Iα∗upup−2u+f(x)u2p−2uinRN,u∈H1RN,$(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

$X=u∈H1RN|∫RNgu2dx<∞$

be equipped with the inner product and norm

$u,v=∫RN∇u∇v+guvdx,u=u,u1/2.$

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

$u,vμ=∫RN∇u∇v+μguvdx,uμ=u,uμ1/2.$

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

$∫RN∇u2+u2dx=∫RN∇u2dx+∫g

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],

$∫RNurdx≤∫g≥cu2dx+∫g(2.1)

where, S the best constant for the embedding of D1,2(ℝN) in L2(ℝN). Moreover, if we assume that u$\begin{array}{}{L}^{\frac{2Np}{N+\alpha }}\left({\mathbb{R}}^{N}\right),\end{array}$ then by the Hardy–Littlewood–Sobolev inequality (see [20, 25, 26]) to the function ∣up$\begin{array}{}{L}^{\frac{2N}{N+\alpha }}\left({\mathbb{R}}^{N}\right),\end{array}$ we obtain, in view of the Hölder inequality and (2.1),

$∫RNIα∗upupdx≤∫RNIα∗up2NN−αdxN−α2N∫RNu2NpN+αdxN+α2N≤CN,α,2NN+α∫RNu2NpN+αdxN+αN$(2.2)

$≤CN,α,2NN+αg(2.3)

where

$CN,α,2NN+α=ΓN−α22απα/2ΓN+α2ΓN2ΓNα/N.$

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

$Jμ,λu=12Aμ,λu−12pBu,$

where

$Aμ,λu:=uμ2−λ∫RNau2dx$

and

$Bu:=−∫RNIα∗upupdx+∫RNfu2pdx.$

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])

$Nμ,λ=u∈X∖0|〈Jμ,λ′u,u〉=0.$

Thus, uNμ,λ if and only if

$Aμ,λu−Bu=0.$

Hence, if uNμ,λ, then

$Jμ,λ(u)=12−12pAμ,λu=12−12pBu.$

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 $\begin{array}{}{h}_{u}^{\prime }\end{array}$(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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+},\phantom{\rule{thinmathspace}{0ex}}{\mathbf{N}}_{\mu ,\lambda }^{-}\text{\hspace{0.17em}and\hspace{0.17em}}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ corresponding to local minima, local maxima and points of inflexion of fibrering maps. We have

$hu′(t)=tAμ,λu−t2p−1Bu$(2.4)

and

$hu″(t)=Aμ,λu−2p−1t2p−2Bu.$

Hence if we define

$Nμ,λ+=u∈Nμ,λ:Aμ,λu−2p−1Bu>0;Nμ,λ0=u∈Nμ,λ:Aμ,λu−2p−1Bu=0;Nμ,λ−=u∈Nμ,λ:Aμ,λu−2p−1Bu<0,$

which indicates that for uNμ,λ, we have $\begin{array}{}{h}_{u}^{\prime }\end{array}$(1) = 0 and u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+},{\mathbf{N}}_{\mu ,\lambda }^{0},{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ if $\begin{array}{}{h}_{u}^{″}\left(1\right)>0,{h}_{u}^{″}\left(1\right)=0,{h}_{u}^{″}\left(1\right)\end{array}$ < 0, respectively. Moreover, it is easy to show that

$Nμ,λ+=u∈Nμ,λ:Aμ,λu−2p−1Bu>0=u∈Nμ,λ:2p−2Bu<0=u∈Nμ,λ:Bu<0.$

Similarly,

$Nμ,λ−=u∈Nμ,λ:Bu>0$

and

$Nμ,λ0=u∈Nμ,λ:Bu=0.$

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

$t(u)=Aμ,λuBu12p−2$(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)u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$. 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)u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$. 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ if and only if Aμ,λ (u),B(u) > 0;

2. a multiple of u lies is $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}.\end{array}$ Then $\begin{array}{}{J}_{\mu ,\lambda }^{\prime }\end{array}$(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

$∫RNIα∗un−upun−updx=∫RNIα∗unpunpdx−∫RNIα∗upupdx+o1.$

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 v0$\begin{array}{}{H}_{0}^{1}\end{array}$ (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

$vn⇀v0 in X,vn→v0 a.e. in RN,vn→v0 in LlocrRN for 2≤r<2∗.$

By Fatou’s Lemma, we have

$∫RNgv02dx≤lim infn→∞∫RNgvn2dx≤lim infn→∞vnμn2μn=0,$

this implies that $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}g{v}_{0}^{2}dx=0\end{array}$ or v0 = 0 a.e. in ℝNΩ and v0$\begin{array}{}{H}_{0}^{1}\end{array}$(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

$∫BNxn,R0vn−v02dx≥d0.$

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

$∫Bxn,R0∩g

Consequently,

$c0≥vnμn2≥μnc∫Bxn,R0∩g≥cvn2dx=μnc∫Bxn,R0∩g≥cvn−v02dx=μnc∫Bxn,R0vn−v02dx−∫Bxn,R0∩g

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 $\begin{array}{}2\le \frac{2Np}{N+\alpha }<{2}^{\ast }.\end{array}$ This completes the proof.□

Next, we consider the following eigenvalue problem

$−Δu(x)+μgxux=λa(x)u(x) forx∈RN.$(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

$Iμu=12∫RN∇u2+μgu2dx,$

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

$λ~1,μa=infu∈X∖0∫RN∇u2+μgu2dx∫RNau2dx.$(2.7)

Then, by (2.1),

$∫RN∇u2+μgu2dx∫RNau2dx≥S2a∞g

this implies that $\begin{array}{}{\stackrel{~}{\lambda }}_{1,\mu }\left(a\right)\ge \frac{{S}^{2}}{{∥a∥}_{\mathrm{\infty }}{\left|\left\{g0.\end{array}$ Moreover, by condition (V3),

$infu∈X∖0∫RN∇u2+μgu2dx∫RNau2dx≤infu∈H01Ω∖0∫RN∇u2+μgu2dx∫RNau2dx=infu∈H01Ω∖0∫Ω∇u2∫Ωa¯Ωu2dx,$

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 $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}a{\phi }_{\mu }^{2}dx=1\end{array}$ such that

$λ~1,μa=∫RN∇φμ2+μgφμ2dx<λ1a¯Ω.$

Furthermore, λ͠1,μ(a) → $\begin{array}{}{\lambda }_{1}^{-}\end{array}$(aΩ) and φμϕ1 as μ → ∞, where ϕ1 is positive principal eigenfunction of problem (1.1).

#### Proof

Let {un} ⊂ X with $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}a{u}_{n}^{2}dx=1\end{array}$ be a minimizing sequence of (2.7), that is

$∫RN∇un2+μgun2dx→λ~1,μa as n→∞.$

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

$un⇀φμ in Xμ,un→φμ a.e. in RN,un→φμ in LlocrRN for 2≤r<2∗.$

Moreover, by condition (V4),

$∫RNaun2dx→∫RNaφμ2dx=1.$

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

$∫RN∇φμ2+μgφμ2dx

which is impossible. Thus, unφμ in Xμ, which implies that $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}a{\phi }_{\mu }^{2}dx=1\text{\hspace{0.17em}and\hspace{0.17em}}{\int }_{{\mathbb{R}}^{N}}{\left|\mathrm{\nabla }{\phi }_{\mu }\right|}^{2}+\mu g{\phi }_{\mu }^{2}dx\end{array}$ = λ͠1,μ(a). Since ∣ φμ∣ ∈ X and

$λ~1,μa=∫RN∇φμ2+μgφμ2dx=∫RN∇φμ2+μgφμ2dx,$

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 $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}a{\phi }_{n}^{2}dx=1\end{array}$ and

$λ~1,μna=∫RN∇φn2+μngφn2dx≤λ1a¯Ω,$

that

$λ~1,μna→d0≤λ1a¯Ω for some d0>0$

and

$φn≤φnμn≤λ1a¯Ω, for n sufficiently large.$

Thus, by Lemma 2.4, we may assume that there exists φ0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that φnφ0 in X and φnφ0 in Lr(ℝN) for all 2 ≤ r < 2. Then

$∫Ω∇φ02dx≤lim infn→∞∫RN∇φn2+μngφn2dx=d0$

and

$limn→∞∫RNaφn2dx=∫Ωa¯Ωφ02dx=1.$

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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$.

#### Lemma 3.1

For each 0 < λ < λ1 (aΩ) there exists μ0(λ) ≥ μ0 with $\begin{array}{}\underset{\lambda \to {\lambda }_{1}^{-}\left({\overline{a}}_{\mathit{\Omega }}\right)}{lim}\end{array}$ μ0(λ) = ∞ such that for every μ > μ0(λ), we have

1. Nμ,λ = $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$;

2. the energy functional Jμ,λ is coercive and bounded below on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$. Furthermore, there exists d0 > 0 such that

$infu∈Nμ,λ−Jμ,λ(u)≥p−1λ~1,μa−λ2pλ1,μad01/p−1>0$(3.1)

for all u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$.

#### 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

$Aμ,λu=uμ2−λ∫RNau2dx≥λ~1,μa−λλ~1,μauμ2>0 for all u∈X∖0.$(3.2)

Thus, by Lemma 2.1, the submanifolds $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ and $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ are empty and so $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }={\mathbf{N}}_{\mu ,\lambda }^{-}.\end{array}$

2. By (2.1) and (3.2), for each μ > μ0(λ) and u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$, we obtain

$λ~1,μa−λλ~1,μauμ2≤Aμ,λu<2p−1Bu≤2p−1f∞g

which indicates that

$uμ≥d0:=Spλ~1,μa−λ2p−1λ~1,μaf∞g

Thus,

$Jμ,λ(u)=p−12pAμ,λu≥p−1λ~1,μa−λ2pλ~1,μad01/p−1>0,$

this implies that the energy functional Jμ,λ is coercive and bounded below on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$. This completes the proof.

We now show that there exists a minimizer on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ which is a critical point of Jμ,λ(u) and so a nontrivial solution of Eq. (Pμ,λ). First, we define

$cλ(Ω)=infu∈Mμ,λ(Ω)Jμ,λ|H01(Ω)(u),$

where

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

Note that

$Jμ,λ|H01Ω(u)=12∫Ω∇u2dx−∫Ωλa¯Ωu2dx−12p−∫ΩIα∗upupdx+∫Ωfu2pdx,$

a restriction of Jμ,λ on $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), and cλ(Ω) independent of μ. Since 0 < λ < λ1(aΩ), similar to the argument of (3.1), we can conclude that $\begin{array}{}{J}_{\mu ,\lambda }{|}_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\end{array}$ is bounded below on Mμ,λ(Ω). Moreover, $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) ⊂ Xμ for all μ > 0, one can see that

$0<η≤infu∈Nμ,λ−Jμ,λ(u)≤cλ(Ω) for all μ≥μ0.$

Taking D0 > cλ(Ω). Then we have

$0<η≤infu∈Nμ,λ−Jμ,λ(u)≤cλ(Ω)(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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ 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} ⊂ $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ such that

$limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u)>0 and Jμ,λ′(un)=o1.$

Since $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}{J}_{\mu ,\lambda }\left(u\right)\end{array}$ < 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, $\begin{array}{}{J}_{\mu ,\lambda }^{\prime }\end{array}$(u0) = 0 and

$un⇀u0 in Xμ,un→u0 a.e. in RN,un→u0 in LlocrRN for 2≤r<2∗.$(3.4)

Then by condition (V4),

$limn→∞∫RNaun2dx=∫RNau02dx.$(3.5)

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

$Bun−u0=Bun−Bu0+o(1).$(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

$∫RNvn2dx≤1μc∫g≥cμgvn2dx+∫g

and

$∫RNvn2pdx≤C01μcvnμ22∗−2p2∗−2∫RN∇vn2dx2∗p−12∗−2+o1≤C01μc2∗−2p2∗−2vnμ2p+o1$

or

$∫RNvn2pdx≤Πμvnμ2p+o(1),$(3.7)

where $\begin{array}{}{\mathit{\Pi }}_{\mu }={C}_{0}{\left(\frac{1}{\mu c}\right)}^{\frac{{2}^{\ast }-2p}{{2}^{\ast }-2}}.\end{array}$ Thus, using (3.4)(3.6) gives

$Jμ,λvn=Jμ,λun−Jμ,λu0+o1 and 〈Jμ,λ′(vn),vn〉=o(1).$(3.8)

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

$D0≥infu∈Nμ,λ−Jμ,λ(u)−Jμ,λu0≥Jμ,λvn−12p〈Jμ,λ′(vn),vn〉+o1≥p−1λ~1,μa−λ2pλ~1,μa∥vn∥μ2+o(1),$

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

$vnμ≤C1+o1 for λ>μ¯0λ.$(3.9)

Since $\begin{array}{}1<\frac{N+\alpha }{N} it follows from (3.5), (3.7) and (3.9) that

$o1=〈Jμ,λ′(vn),vn〉≥vnμ21−f∞Πμvnμ2p−2+o(1)≥vnμ21−f∞ΠμC12p−2+o1.$(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

$Jμ,λ(u0)=limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u),$

which indicates that u0 is a minimizer on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$. This completes the proof.□

We are now ready to prove Theorem 1.1: By Theorem 3.2, Jμ,λ has a minimizer u0 on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ for all μ > μ͠0(λ). Since B(u0) > 0 and u0$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$, 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 λ$\begin{array}{}{\lambda }_{1}^{-}\left({\overline{a}}_{\mathrm{\Omega }}\right)\end{array}$ 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

$limλ→λ1−a¯Ωinfu∈Nμ,λ−Jμ,λ(u)=0.$

#### 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)ϕ1$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$, where

$t(ϕ1)=∫RN(|∇ϕ1|2dx−λaϕ12)dxBϕ11/2p−2=(λ1a¯Ω−λ)∫RNaϕ12dxBϕ11/2p−2>0.$

Thus,

$Jμ,λ(t(ϕ1)ϕ1)=p−22p(λ1a¯Ω−λ)∫RNaϕ12dxp/p−1Bϕ11/p−1→0 as λ→λ1−a¯Ω.$

Since 0 < $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ Jμ,λ(u) ≤ Jμ,λ(t(ϕ1)ϕ1), it follows that $\begin{array}{}\underset{\lambda \to {\lambda }_{1}^{-}\left(a\right)}{lim}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}{J}_{\mu ,\lambda }\left(u\right)\end{array}$ = 0. This completes the proof. □

Next, we are ready to prove Theorem 1.2:

1. Since λn$\begin{array}{}{\lambda }_{1}^{-}\left({\overline{a}}_{\mathit{\Omega }}\right)\end{array}$ 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 = $\begin{array}{}\frac{{u}_{n}}{\parallel {u}_{n}{\parallel }_{{\mu }_{n}}}\end{array}$. Since ∥vnμn = 1, by Lemma 2.4, there exist subsequence {vn} and v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that vnv0 in Lr(ℝN) for 2 ≤ r < 2 and B(vn) → B(v0). Hence

$limn→∞∫RNavn2dx=∫RNav02dx.$

By Theorem 3.3,

$Jμn,λn(un)=p−22punμn2−λn∫RNaun2dx=p−22pBun→0 as n→∞,$

dividing by $\begin{array}{}\parallel {u}_{n}{\parallel }_{{\mu }_{n}}^{2}\end{array}$ it is easy to see that

$limn→∞vnμn2−λn∫RNavn2dx=0$

and

$limn→∞∥un∥μn2p−2Bvn=0.$

Thus,

$limn→∞λn∫RNavn2dx=λ1a¯Ω∫RNav02dx=1$

and

$limn→∞Bvn=Bv0=0.$

Now, we show that

$limn→∞∫RN|∇vn|2dx=∫RN|∇v0|2dx.$

If not, then we may assume that

$0≤∫RN(|∇v0|2−λ1a¯Ωav02)dx

which is impossible. Thus, we must have

$∫Ω(|∇v0|2−λ1a¯Ωav02)dx=limn→∞vnμn2−λn∫RNavn2dx=0,$

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 $\begin{array}{}{\int }_{{\mathbb{R}}^{N}}a{v}_{0}^{2}dx\ne 0\end{array}$, this is impossible. Hence {un} is bounded. By Lemma 2.4, we may assume that there exists u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that

$limn→∞∫RNaun2dx=∫RNau02dx and limn→∞Bun=Bu0.$

Moreover, by Theorem 3.3,

$Jμn,λn(un)=p−12pAμn,λnun=p−22pBun→0 as n→∞,$

which indicates that

$limn→∞Bun=Bu0=0.$

Since

$0≤∫RN(|∇u0|2−λ1a¯Ωau02)dx≤lim infn→∞Aμn,λnun=0,$

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 u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu0 in X and unu0 in Lr(ℝN) for all 2 ≤ r < 2. Now for any φ$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(Ω), because $\begin{array}{}〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}\right),\phi 〉\end{array}$ = 0, it is easy to check that

$∫Ω∇u0∇φdx=λ∫Ωa¯Ωu0φdx+∫Ωfu0p−2u0φdx,$

that is, u0 is a weak solution of Eq. (P) by the density of $\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(Ω) in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω). Now, we show that unu0 in X. Because $\begin{array}{}〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}\right),{u}_{n}〉=〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}\right),{u}_{0}〉\end{array}$ = 0, we have

$unμn2=λ∫RNaun2dx+∫RNfunpdx$(3.11)

and

$un,u0μn=λ∫RNaunu0dx+∫RNfunp−2unu0dx.$(3.12)

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

$limn→∞unμn2=limn→∞un,u0μn=limn→∞un,u0=u02.$

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

$u02≤lim infn→∞un2≤limn→∞unμn2,$

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

$Aμ,λϕ1=∫RN(|∇ϕ1|2−λaϕ12)dx=(λ1a¯Ω−λ)∫RNaϕ12dx<0 for all μ>0.$

Hence, if B(ϕ1) < 0, then by Lemma 2.1, $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ ≠ ∅. 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 $\begin{array}{}{∥u∥}_{\mu }^{2}\le \lambda {\int }_{{\mathbb{R}}^{N}}a{u}^{2}dx\end{array}$ 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}=\mathrm{\varnothing }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathbf{N}}_{\mu ,\lambda }={\mathbf{N}}_{\mu ,\lambda }^{+}\cup {\mathbf{N}}_{\mu ,\lambda }^{-}\right)\end{array}$ is an important condition for establishing the existence of minimizers.

Let

$Aμ,λ=u∈X∖0:Aμ,λu≤0$

and

$Bμ,λ=u∈X∖0:Bu≥0.$

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, $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ = ∅ 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 λn$\begin{array}{}{\lambda }_{1}^{+}\end{array}$(aΩ) and μn → ∞ such that

$Aμn,λnwn=∥wn∥μn2−λn∫RNawn2dx≤0$

and

$Bwn=−∫RNIα∗wnpwnpdx+∫RNf|wn|2pdx≥0.$

Let un = $\begin{array}{}\frac{{w}_{n}}{\parallel {w}_{n}{\parallel }_{{\mu }_{n}}}\end{array}$. Since ∥un∥ ≤ ∥unμn = 1, by Lemma 2.4, we may assume that there exists u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu0 a.e. in ℝN, unu0 in Lr(ℝN) for all 2 ≤ r < 2 and B(un) → B(u0). Then

$limn→∞λn∫RNaun2dx=λ1+a¯Ω∫RNau02dx≥1.$(4.1)

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

$∫Ω|∇u0|2−λ1a¯Ωa¯Ωu02dx=∫RN|∇u0|2−λ1a¯Ωau02dx

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

$(I)∫Ω(|∇u0|2−λ1a¯Ωa¯Ωu02)dx≤0,(II)Bu0≥0.$

But (I) implies that u0 = 1 for some k and then (II) implies that k = 0 which is impossible as $\begin{array}{}{\lambda }_{1}^{+}\end{array}$(aΩ) ∫N $\begin{array}{}a{u}_{0}^{2}dx\end{array}$ ≥ 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ ≠ ∅, then there exists u0$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ such that u0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Therefore, $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ = ∅ for all λ1(aΩ) < λ < λ1(aΩ) + δ0 and μ > μ̂0. This completes the proof. □

When $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ = ∅, any non-zero minimizer for Jμ,λ on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ (or on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$) 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0}\end{array}$ = ∅, it is possible to obtain more information about the nature of the Nehari manifold. Since B(ϕ1) < 0, we can obtain that $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ ≠ ∅ 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. $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ is uniform bounded.

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

$κ1≤infu∈Nμ,λ+Jμ,λ(u)<κ2.$

#### Proof

1. Suppose on the contrary. Then there exist sequences $\begin{array}{}\left\{{\mu }_{n}\right\}\subset {\mathbb{R}}_{+}^{N}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{{u}_{n}\right\}\subset {\mathbf{N}}_{{\mu }_{n},\lambda }^{+}\end{array}$ such that μn → ∞ and ∥unμn → ∞ as n → ∞. Clearly,

$Aμn,λun=Bun<0.$(4.2)

Let vn = $\begin{array}{}\frac{{u}_{n}}{\parallel {u}_{n}{\parallel }_{{\mu }_{n}}}\end{array}$. Then by Lemma 2.4, we may assume that there exists v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that

$vn⇀v0 in X;vn→v0 in LrRN for all 2≤r<2∗,$

and

$limn→∞Bvn=Bv0.$(4.3)

Thus,

$limn→∞∫RNavn2dx=∫RNav02dx.$(4.4)

Moreover, by Fatou’s Lemma,

$∫RN|∇v0|2dx≤lim infn→∞∫RN|∇vn|2dx.$(4.5)

Dividing (4.2) by $\begin{array}{}\parallel {u}_{n}{\parallel }_{{\mu }_{n}}^{2}\end{array}$ gives

$Aμn,λvn=∥un∥μnp−2Bvn<0.$(4.6)

Since

$limn→∞Aμn,λvn=1−λlimn→∞∫RNavn2dx=1−λ∫RNav02dx$

and ∥unμn → ∞, it obtain that B(v0) = 0 and ∫N $\begin{array}{}a{v}_{0}^{2}dx\end{array}$ > 0 from the conclusions (4.3) and (4.6). Thus, v0 ∈ 𝓑μ,λ for all μ > 0. Moreover, by v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω), (4.5) and (4.4), for every μ > 0,

$v0μ2−λ∫RNav02dx=∫RN|∇v0|2−λav02dx

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

$λv0μ2−∫RNav02dx=∫RN|∇v0|2−λav02dx

since ∫N $\begin{array}{}g{v}_{0}^{2}dx\end{array}$ = 0. Hence v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Since vnv0 in Xμ, then ∥v0μ = 1. Hence v0 ∈ 𝓑μ,λ. Moreover,

$v0μ2−λ∫RNav02dx=limn→∞Aμn,λvn≤0$

and so v0 ∈ 𝓐μ,λ. Thus, v0 ∈ 𝓐μ,λ ∩ 𝓑μ,λ which is impossible. Hence $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ is uniform bounded for μ > 0 sufficiently large.

2. By part (i), there exists C0 > 0 such that ∥uμC0 for all u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$. Hence, making use of (2.1), for u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ we have

$Jμ,λ(u)=p−12pBu≥−p−12p∫RNIα∗upupdx+f∞∫RNu2pdx≥−p−12pC1uμ2p≥−p−12pSpC1C0p=κ1.$(4.7)

Moreover, by B(ϕ1) < 0 and ∫Ω|∇ ϕ1|2dxλΩ $\begin{array}{}a{\varphi }_{1}^{2}dx\end{array}$ < 0, which indicates that the function hϕ1(t) = Jμ,λ(1) have $\begin{array}{}{t}_{0}^{+}\end{array}$ > 0 and κ2 < 0 are independent of μ such that $\begin{array}{}{t}_{0}^{+}\phi \in {\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ and

$inf0

This implies that

$infu∈Nμ,λ+Jμ,λu≤κ2<0 for all μ>maxμ¯1,μ¯2.$(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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$.

#### Proof

By Lemmas 4.1, 4.2 and the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$ such that

$limn→∞Jμ,λ(un)=infu∈Nμ,λ+Jμ,λ(u)≤κ2 and Jμ,λ′(un)=o1$

and there exists C0 > 0 such that ∥unμC0. Thus, there exist a subsequence {un} and u0Xμ such that $\begin{array}{}{J}_{\mu ,\lambda }^{\mathrm{\prime }}\end{array}$ (u0) = 0 and

$un⇀u0 in Xμ,un→u0 a.e. in RN,un→u0 in LlocrRN for 2≤r<2∗.$

Then by condition (V4),

$limn→∞∫RNaun2dx=∫RNau02dx.$(4.9)

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

$Bun−u0=Bun−Bu0+o(1).$(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

$∫RNvn2dx≤1μc∫g≥cμgvn2dx+∫g

and

$∫RNvn2pdx≤C01μcvnμ22∗−2p2∗−2∫RN∇vn2dx2∗p−12∗−2+o1≤C01μc2∗−2p2∗−2vnμ2p+o1$

or

$∫RNvn2pdx≤Πμvnμ2p+o(1),$(4.11)

where $\begin{array}{}{\mathit{\Pi }}_{\mu }={C}_{0}{\left(\frac{1}{\mu c}\right)}^{\frac{{2}^{\ast }-2p}{{2}^{\ast }-2}}\end{array}$. Thus, using (4.9) and (4.10) gives

$Jμ,λvn=Jμ,λun−Jμ,λu0+o1 and Jμ,λ′(vn),vn=o(1).$(4.12)

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

$κ2+κ1≥infu∈Nμ,λ+Jμ,λ(u)−Jμ,λu0≥Jμ,λvn−12pJμ,λ′(vn),vn+o1≥p−12p∥vn∥μ2+o(1),$

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

$vnμ≤C1+o1 for μ>0 sufficiently large.$(4.13)

Since 1 < $\begin{array}{}\frac{N+\alpha }{N}, it follows from (4.9), (4.11) and (4.13) that

$o1=Jλ′(vn),vn≥vnμ21−C0Πμvnμ2p−2+o(1)≥vnμ21−f∞ΠμC12p−2+o1.$(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

$Jμ,λ(u0)=limn→∞Jμ,λ(un)=infu∈Nμ,λ+Jμ,λ(u)≤κ0<0,$

which implies that u0 is a minimizer on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$. □

We now turn our attention to $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$.

#### 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 u$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$;

2. every minimizing sequence for Jμ,λ(u) on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ is bounded;

3. $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ Jμ,λ(u) > 0.

#### Proof

1. Suppose on the contrary. Then there exist {μn} ⊂ ℝ+ and {un} ⊂ $\begin{array}{}{\mathbf{N}}_{{\mu }_{n},\lambda }^{-}\end{array}$ such that μn → ∞ and ∥unμn → 0. Hence, by (2.1),

$0

Let vn = $\begin{array}{}\frac{{u}_{n}}{\parallel {u}_{n}{\parallel }_{{\mu }_{n}}}\end{array}$. Then, by Lemma 2.4, there exist subsequence {vn} and v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that

$vn⇀v0 in X; vn→v0 in LrRN for all 2≤r<2∗.$

Thus,

$limn→∞∫RNavn2dx=∫RNav02dx$(4.15)

and

$Aμn,λvn=unμn2p−2Bvn→0 as n→∞.$(4.16)

Moreover, by (4.15), (4.16), v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) and Fatou’s Lemma, we can obtain that

$0=limn→∞Aμn,λvn=1−λlimn→∞∫RNavn2dx=1−λ∫RNav02dx,$

and for every μ > 0

$v0μ2−∫RNλav02dx=∫RN|∇v0|2−λav02dx≤lim infn→∞vnμn2−∫RNλavn2dx=0,$

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 $\begin{array}{}\frac{{v}_{0}}{\parallel {v}_{0}{\parallel }_{\mu }}\end{array}$ ∈ 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 $\begin{array}{}{\mathbf{N}}_{{\mu }_{n},\lambda }^{-}\end{array}$ is unbounded for all n, that is for every n there exists a minimizing sequence {un,m} ⊂ $\begin{array}{}{\mathbf{N}}_{{\mu }_{n},\lambda }^{-}\end{array}$ such that ∥un,mμn → ∞ as m → ∞. Moreover,

$Aμn,λun,m=Bun,m→p−12pinfu∈Nμn,λ−Jμn,λ(u) as m→∞,$(4.17)

where $\begin{array}{}\underset{u\in {\mathbf{N}}_{{\mu }_{n},\lambda }^{-}}{inf}\end{array}$ Jμn,λ(u) ≥ 0 for all n. Let wn = un,n. Then wn$\begin{array}{}{\mathbf{N}}_{{\mu }_{n},\lambda }^{-}\end{array}$ and ∥wnμn → ∞ as n → ∞. Let vn = $\begin{array}{}\frac{{w}_{n}}{\parallel {w}_{n}{\parallel }_{{\mu }_{n}}}\end{array}$. Then by Lemma 2.4, we may assume that there exist subsequence {vn} and v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that vnv0 in X, vnv0 in Lr(ℝN) for all 2 ≤ r < 2 and B(vn) → B(v0). Then by condition (V4)

$limn→∞∫RNavn2dx=∫RNav02dx.$(4.18)

Dividing (4.17) by $\begin{array}{}\parallel {w}_{n}{\parallel }_{{\mu }_{n}}^{2}\end{array}$ and m = n gives

$Aμn,λvn=∥wn∥μnp−2Bvn→0.$(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|2−λav02dx=∥v0∥2−λ∫RNav02dx

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, ∫N$\begin{array}{}V{v}_{0}^{2}dx\end{array}$ = 0 and

$∥v0∥μ2−λ∫RNav02dx=Bv0=0.$

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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ is bounded for μ sufficiently large.

3. Assume that $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ Jμ,λ(u) = 0. Then by the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ such that

$limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u) and Jμ,λ′(un)=o1.$

By part (ii), {un} is bounded and so there exist a subsequence {un} and u0Xμ such that $\begin{array}{}{J}_{\mu ,\lambda }^{\mathrm{\prime }}\end{array}$(u0) = 0 and

$un⇀u0 in Xμ,un→u0 a.e. in RN,un→u0 in LlocrRN for 2≤r<2∗.$

Then by condition (V4)

$limn→∞∫RNavn2dx=∫RNav02dx.$(4.20)

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

$Bun−u0=Bun−Bu0+o(1).$(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

$RNvn2dx≤1μc∫g≥cμgvn2dx+∫g

and

$∫RNvn2pdx≤C01μcvnμ22∗−2p2∗−2∫RN∇vn2dx2∗p−12∗−2+o1≤C01μc2∗−2p2∗−2vnμ2p+o1.$

or

$∫RNvn2pdx≤Πμvnμ2p+o(1),$(4.22)

where $\begin{array}{}{\mathit{\Pi }}_{\mu }={C}_{0}{\left(\frac{1}{\mu b}\right)}^{\frac{{2}^{\ast }-2p}{{2}^{\ast }-2}}{S}^{-N\left(p-1\right)}\end{array}$. Thus, using (4.21) and unu0 in Xμ gives

$Jμ,λvn=Jμ,λun−Jμ,λu0+o1 and Jμ,λ′(vn),vn=o(1).$(4.23)

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

$infu∈Nμ,λ−Jμ,λ(u)−Jμ,λu0≥Jμ,λvn−12pJμ,λ′(vn),vn+o1≥p−12p∥vn∥μ2+o(1).$(4.24)

Suppose that $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ Jμ,λ(u) = 0.

(iiiA) If u0$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$, then by (4.24) and u0 = 0, $\begin{array}{}\parallel {v}_{n}{\parallel }_{\mu }^{2}\end{array}$ → 0, this shows that unu0 in Xμ, and so

$Jμ,λ(u0)=limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u)=0.$

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 u0$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{+}\end{array}$, then by (4.7) and (4.24), there exists C0 > 0 such that

$vnμ≤C0+o1 for μ enough large.$(4.25)

Since 1 < $\begin{array}{}\frac{N+\alpha }{N}, it follows from (4.18), (4.22) and (4.25) that

$o1=Jμ,λ′(vn),vn≥vnμ21−f∞Πμvnμ2p−2+o(1)≥vnμ21−f∞ΠμC02p−2+o1.$(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 u0$\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ this is a contradiction. Thus, $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ 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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$.

#### Proof

By Lemmas 4.1, 4.4 (iii) and the Ekeland variational principle [19], there exists a minimizing sequence {un} ⊂ $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ such that

$limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u) and Jμ,λ′(un)=o1.$

Similar the argument in (3.3), there exists D0 > 0 independent of μ such that $\begin{array}{}\underset{u\in {\mathbf{N}}_{\mu ,\lambda }^{-}}{inf}\end{array}$ 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 $\begin{array}{}{J}_{\mu ,\lambda }^{\mathrm{\prime }}\end{array}$(u0) = 0 and

$un⇀u0 in Xμ,un→u0 a.e. in RN,un→u0 in LlocrRN for 2≤r<2∗.$

Then by condition (V4),

$limn→∞∫RNavn2dx=∫RNav02dx,$(4.27)

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

$Bun−u0=Bun−Bu0+o(1).$(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

$∫RNvn2dx≤1μc∫g≥cμgvn2dx+∫g

and

$∫RNvn2pdx≤C01μcvnμ22∗−2p2∗−2∫RN∇vn2dx2∗p−12∗−2+o1≤C01μc2∗−2p2∗−2vnμ2p+o1.$

or

$∫RNvn2pdx≤Πμvnμ2p+o(1),$(4.29)

where $\begin{array}{}{\mathit{\Pi }}_{\mu }={C}_{0}{\left(\frac{1}{\mu c}\right)}^{\frac{{2}^{\ast }-2p}{{2}^{\ast }-2}}.\end{array}$ Thus, using (4.27) and (4.28) gives

$Jμ,λvn=Jμ,λun−Jμ,λu0+o1 and Jμ,λ′vn=o(1).$(4.30)

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

$D0+κ1≥infu∈Nμ,λ−Jμ,λ(u)−Jμ,λu0≥Jμ,λvn−1p〈Jμ,λ′(vn),vn〉+o1≥p−22p∥vn∥μ2+o(1),$(4.31)

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

$vnμ≤C1+o1.$(4.32)

Since $\begin{array}{}1<\frac{N+\alpha }{N} it follows from (4.29) (4.31) and (4.32) that

$o1=Jλ′(vn),vn≥vnμ21−f∞Πμvnμ2p−2+o(1)≥vnμ21−f∞ΠμC12p−2+o1.$(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

$Jμ,λ(u0)=limn→∞Jμ,λ(un)=infu∈Nμ,λ−Jμ,λ(u),$

which implies that u0 is a minimizer on $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$.□

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 $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{\left(1\right)}\text{\hspace{0.17em}and\hspace{0.17em}}{\mathbf{N}}_{\mu ,\lambda }^{\left(2\right)},\end{array}$ that is there exist $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\in {\mathbf{N}}_{\mu ,\lambda }^{+}\text{\hspace{0.17em}and\hspace{0.17em}}{u}_{\mu ,\lambda }^{\left(2\right)}\in {\mathbf{N}}_{\mu ,\lambda }^{-}\end{array}$ such that

$Jμ,λ(uμ,λ1)=infu∈Nμ,λ+Jμ,λ(u)<κ2<0

Since $\begin{array}{}{J}_{\mu ,\lambda }\left({u}_{\mu ,\lambda }^{\left(j\right)}\right)={J}_{\mu ,\lambda }\left(|{u}_{\mu ,\lambda }^{\left(j\right)}|\right)\end{array}$ for j = 1, 2, we may assume that these minimizers are positive. Moreover, by Lemma 4.1, $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }={\mathbf{N}}_{\mu ,\lambda }^{+}\cup {\mathbf{N}}_{\mu ,\lambda }^{-}.\end{array}$ It follows that the minimizers are local minimizers in Nμ,λ which do not lie in $\begin{array}{}{\mathbf{N}}_{\mu ,\lambda }^{0},\end{array}$ and so by Lemma 2.2, $\begin{array}{}{u}_{\mu ,\lambda }^{\left(1\right)}\text{\hspace{0.17em}and\hspace{0.17em}}{u}_{\mu ,\lambda }^{\left(2\right)}\end{array}$ are positive solutions of Eq. (Pμ,λ). This completes the proof.

Next, we are ready to prove Theorem 1.4: (i) Since $\begin{array}{}{\mathbf{N}}_{{\mu }_{n},{\lambda }_{n}}^{+}\end{array}$ is uniformly bounded, then {un} is bounded, from Lemma 2.4, we may assume that there exists u0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) such that unu0 in X, unu0 in Lr(ℝN) for all 2 ≤ r < 2 and B(un) → B(u0). We also have

$limn→∞∫RNaun2dx=∫RNau02dx$

and

$un2−λn∫RNaun2dx≤Aμn,λnun=Bun<0 for n sufficiently large.$

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

$∫RN(|∇u0|2−λ1a¯Ωau02)dx=∫RN(|∇u0|2+Vu02−λ1a¯Ωau02)dx

which is impossible. Thus, unu0 in X and so

$∫Ω(|∇u0|2−λ1a¯Ωau02)dx≤∫RN(|∇u0|2+Vu02−λ1a¯Ωau02)dx=Bu0≤0,$

this implies that $\begin{array}{}{\int }_{\mathit{\Omega }}\left(|\mathrm{\nabla }{u}_{0}{|}^{2}-{\lambda }_{1}\left({\overline{a}}_{\mathit{\Omega }}\right){\overline{a}}_{\mathit{\Omega }}{u}_{0}^{2}\right)dx=0\end{array}$ 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 $\begin{array}{}{v}_{n}=\frac{{u}_{n}}{\parallel {u}_{n}{\parallel }_{{\mu }_{n}}}.\end{array}$ Then by Lemma 2.4, we may assume that there exists v0$\begin{array}{}{H}_{0}^{1}\end{array}$(Ω) \ {0} such that vnv0 in X, vnv0 in Lr(ℝN) for all 2 ≤ r < 2 and B(vn) → B(v0). Thus,

$limn→∞∫RNavn2dx=∫RNav02dx.$(4.34)

Clearly,

$vn2−λn∫RNavn2dx≤Aμn,λnvn=∥un∥μn2p−2Bvn<0$(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|2−λ1a¯Ωa¯Ωv02)dx=∫RN(|∇u0|2+Vv02−λ1a¯Ωa¯Ωv02)dx

and so

$∫Ω(|∇v0|2−λ1a¯Ωa¯Ωv02)dx<0,$

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

$∫Ω(|∇v0|2−λ1a¯Ωa¯Ωv02)dx=0.$

Therefore, v0 = ϕ1.

(ii) For any sequence μn → ∞, let $\begin{array}{}{u}_{n}^{\left(j\right)}:={u}_{{\mu }_{n}}^{\left(j\right)}\end{array}$ (j = 1, 2) be the solutions obtained in Theorem 1.3 with $\begin{array}{}{u}_{{\mu }_{n}}^{\left(1\right)}\end{array}$$\begin{array}{}{\mathbf{N}}_{{\mu }_{n},\lambda }^{+}\text{\hspace{0.17em}and\hspace{0.17em}}{u}_{{\mu }_{n}}^{\left(2\right)}\in {\mathbf{N}}_{{\mu }_{n},\lambda }^{-}.\end{array}$ 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

$unjμn≤c0.$(4.36)

Therefore, by Lemma 2.4, we may assume that there exist $\begin{array}{}{u}_{0}^{\left(j\right)}\in {H}_{0}^{1}\left(\mathit{\Omega }\right)\end{array}$ such that $\begin{array}{}{u}_{n}^{\left(j\right)}⇀{u}_{0}^{\left(j\right)}\end{array}$ in X and $\begin{array}{}B\left({u}_{n}^{\left(j\right)}\right)\to B\left({u}_{0}^{\left(j\right)}\right).\end{array}$ Now for any φ$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$ (Ω), because $\begin{array}{}〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}^{\left(j\right)}\right),\phi 〉=0,\end{array}$ it is easy to check that

$∫Ω∇u0j∇φdx−λ∫Ωa¯Ωu0jφdx+∫ΩIα∗u0jpu0jp−2u0jφdx=∫Ωfu0j2p−2u0jφdx,$

that is, $\begin{array}{}{u}_{0}^{\left(j\right)}\end{array}$ are weak solutions of Eq. (P) by the density of $\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(Ω) in $\begin{array}{}{H}_{0}^{1}\end{array}$(Ω). Now, we show that $\begin{array}{}{u}_{n}^{\left(j\right)}\to {u}_{0}^{\left(j\right)}\end{array}$ in X for j = 1, 2. Because $\begin{array}{}〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}^{\left(j\right)}\right),{u}_{n}^{\left(j\right)}〉=〈{J}_{{\mu }_{n},\lambda }^{\mathrm{\prime }}\left({u}_{n}^{\left(j\right)}\right),{u}_{0}^{\left(j\right)}〉=0,\end{array}$ we have

$unjμn2−λ∫RNaunj2dx+∫ΩIα∗unjpunjpdx=∫RNfunjpdx$(4.37)

and

$unj,u0jμn−λ∫RNaunju0jdx+∫ΩIα∗unjpunjp−2unju0jdx=∫RNfunjp−2unju0jdx.$(4.38)

By (4.36)(4.38) and $\begin{array}{}{u}_{n}^{\left(j\right)}\to {u}_{0}^{\left(j\right)}\end{array}$ in Lr(ℝN) for all 2 ≤ r < 2, we have

$limn→∞unjμn2=limn→∞unj,u0jμn=limn→∞unj,u0j=u0j2.$

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

$u0j2≤lim infn→∞unj2≤limn→∞unjμn2,$

and thus, $\begin{array}{}{u}_{n}^{\left(j\right)}\to {u}_{0}^{\left(j\right)}\end{array}$ in X for j = 1, 2. By Lemma 4.4 (i) and the fact that $\begin{array}{}{∥{u}_{n}^{\left(2\right)}∥}_{{\mu }_{n}}>{c}_{0}>0,\end{array}$ which implies that $\begin{array}{}{u}_{0}^{\left(2\right)}\ne 0.\end{array}$ Moreover, by (4.8), there exists κ0 < 0

$Jμn,λun1=infu∈Nμn,λ+Jμn,λu≤κ0 for all n.$

Thus,

$Jμ,λ|H01Ωu01≤κ0<0,$

which implies that $\begin{array}{}{u}_{0}^{\left(1\right)}\ne 0.\end{array}$ To complete the proof, it remains to show that $\begin{array}{}{u}_{0}^{\left(1\right)}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{u}_{0}^{\left(2\right)}\end{array}$ are distinct. That $\begin{array}{}{J}_{\mu ,\lambda }{|}_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\left({u}_{0}^{\left(1\right)}\right)\le {\kappa }_{0}<0\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{J}_{\mu ,\lambda }{|}_{{H}_{0}^{1}\left(\mathit{\Omega }\right)}\left({u}_{0}^{\left(2\right)}\right)>0\end{array}$ implies that $\begin{array}{}{u}_{0}^{\left(1\right)}\ne {u}_{0}^{\left(2\right)}.\end{array}$ 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.

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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,

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