The effect of nonlocal term on the superlinear Kirchhoff type equations in $\mathbb{R}^{N}$

TL;DR

This paper investigates how the nonlocal term influences the existence and multiplicity of positive solutions in superlinear Kirchhoff type equations in , employing variational methods and inequalities.

Contribution

It introduces a novel analysis of the effects of functions m and q on solutions, extending previous work on Kirchhoff equations with new conditions.

Findings

Established existence of positive solutions under certain conditions.

Proved multiplicity of solutions using minimax methods.

Demonstrated the influence of m and q functions on solution behavior.

Abstract

We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right) u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying $\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $x\in \mathbb{R}^{N}$ for any $2<k<2^{\ast}$($2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.

Equations

$$\left\{\begin{array}[]{ll}-M\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)\Delta u+\lambda V\left(x\right)u=f(x,u)&\text{in }\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),&\end{array}\right.$$

$$u_{tt}-\left(a{\int }_{\Omega }|\nabla u{|}^{2}dx+b\right)\Delta u=f(x,u),$$

$$\left\{\begin{array}[]{ll}-M\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)\Delta u=f(x,u)&\text{in }\Omega,\\ u\in H_{0}^{1}(\Omega).&\end{array}\right.$$

$$\left\{\begin{array}[]{ll}-\varepsilon^{2}M\left(\varepsilon^{2-N}\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)\Delta u+V\left(x\right)u=f(u)&\text{in }\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}).&\end{array}\right.$$

$${\int }_{\sigma }^{\eta }m(t)dt\ge \frac{2(\eta -\sigma )}{k}m(\eta )$$

$$\underset{s\to \infty }{\lim }\frac{f\left(x,s\right)}{s^{k-1}}=q\left(x\right)\text{ uniformly in }x\in {\mathbb{R}}^N;$$

$${\int }_{\sigma }^{\eta }m(t)dt$$

$$\underset{u\in H^1({\mathbb{R}}^N)}{\inf }\frac{{\left({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)}^{k/2}}{{\int }_{{\mathbb{R}}^N}q(x)|u{|}^{k}dx}=0.$$

$${\overline{\nu }}_1^{(k)}:=\underset{u\in D^{1,2}({\mathbb{R}}^N)}{\inf }\frac{{\left({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)}^{k/2}}{{\int }_{{\mathbb{R}}^N}|x{|}^{\frac{k(N-2)}{2}-N}|u{|}^{k}dx}>0$$

$$\underset{s\to \infty }{\lim }\frac{f\left(x,s\right)}{s^{k-1}}=q\left(x\right)\text{ uniformly in }x\in {\mathbb{R}}^N.$$

$${\overline{\mu }}_1^{(k)}:=\underset{u\in H^1({\mathbb{R}}^N)}{\inf }\frac{{\left({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)}^{k/2}}{{\int }_{{\mathbb{R}}^N}q(x)|u{|}^{k}dx}\ge \frac{{\overline{\nu }}_1^{(k)}}{c_{\ast }}>0.$$

$$X=\left\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V\left(x\right)u^{2}dx<\infty \right\}$$

$$⟨u,v⟩={\int }_{{\mathbb{R}}^N}\left(\nabla u\nabla v+V(x)uv\right)dx,\Vert u\Vert ={⟨u,u⟩}^{1/2}.$$

$${⟨u,v⟩}_{\lambda }={\int }_{{\mathbb{R}}^N}\left(\nabla u\nabla v+\lambda V(x)uv\right)dx,{\Vert u\Vert }_{\lambda }={⟨u,u⟩}_{\lambda }^{1/2}.$$

$${\int }_{{\mathbb{R}}^N}{u}^{2}dx$$

$${\int }_{{\mathbb{R}}^N}{u}^{2}dx\le \frac{4}{N{c}_0}{\int }_{{\mathbb{R}}^N}V\left(x\right)u^{2}dx+A_N^{8/N}{\left|\left\{V<c_0\right\}\right|}^{2/N}{\Vert u\Vert }_{D^{1,2}}^2.$$

$${\Vert u\Vert }_{H^1}^2\le \max \left\{1+A_N^{8/N}{\left|\left\{V<c_0\right\}\right|}^{2/N},\frac{4}{N{c}_0}\right\}{\Vert u\Vert }^2.$$

$${\Vert u\Vert }_{H^1}^2\le \left(1+A_N^{8/N}{\left|\left\{V<c_0\right\}\right|}^{2/N}\right){\Vert u\Vert }_{\lambda }^2\text{ for }\lambda \ge \frac{4}{N{c}_0}{\left(1+A_N^{8/N}{\left|\left\{V<c_0\right\}\right|}^{2/N}\right)}^{-1}.$$

$${\Vert u\Vert }_{H^1}^2\le \max \left\{1+S^2{\left|\left\{V<c_0\right\}\right|}^{2/N},c_0^{-1}\right\}{\Vert u\Vert }^2$$

$${\Vert u\Vert }_{H^1}^2\le \left(1+S^2{\left|\left\{V<c_0\right\}\right|}^{2/N}\right){\Vert u\Vert }_{\lambda }^2\text{ for }\lambda \ge c_0^{-1}\left(1+S^2{\left|\left\{V<c_0\right\}\right|}^{2/N}\right),$$

$$\alpha_{N}:=\left\{\begin{array}[]{ll}\max\left\{1+A_{N}^{8/N}\left|\left\{V<c_{0}\right\}\right|^{2/N},\frac{4}{Nc_{0}}\right\}&\text{ for }N=1,2;\\ \max\left\{1+S^{2}\left|\left\{V<c_{0}\right\}\right|^{2/N},c_{0}^{-1}\right\}&\text{ for }N\geq 3.\end{array}\right.$$

$${\Vert u\Vert }_{H^1}^2\le {\alpha }_N{\Vert u\Vert }^2,$$

$${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{r}dx\le S_r^{-r}{\Vert u\Vert }_{H^1}^r\le S_r^{-r}{\left(1+A_N^{8/N}{\left|\left\{V<c_0\right\}\right|}^{2/N}\right)}^{r/2}{\Vert u\Vert }_{\lambda }^r$$

$${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{r}dx$$

$$\Theta_{r,N}:=\left\{\begin{array}[]{ll}S_{r}^{-r}\left(1+A_{N}^{8/N}\left|\left\{V<c_{0}\right\}\right|^{2/N}\right)^{r/2}&\text{ if }N=1,2;\\ S^{-r}\left|\left\{V<c_{0}\right\}\right|^{\frac{2^{\ast}-r}{2^{\ast}}}&\text{ if }N\geq 3,\end{array}\right.$$

$$\Lambda_{N}:=\left\{\begin{array}[]{ll}\frac{4}{Nc_{0}}\left(1+A_{N}^{8/N}\left|\left\{V<c_{0}\right\}\right|^{2/N}\right)^{-1}&\text{ if }N=1,2;\\ c_{0}^{-1}S^{2}\left|\left\{V<c_{0}\right\}\right|^{-2/N}&\text{ if }N\geq 3.\end{array}\right.$$

$${\int }_{{\mathbb{R}}^N}{\left|u\right|}^{r}dx\le {\Theta }_{r,N}{\Vert u\Vert }_{\lambda }^r.$$

$$J_{\lambda ,a}\left(u\right)=\frac{a}{2}\hat{m}\left({\Vert u\Vert }_{D^{1,2}}^2\right)+\frac{1}{2}\left(b{\Vert u\Vert }_{D^{1,2}}^2+{\int }_{{\mathbb{R}}^N}\lambda V(x)u^{2}dx\right)-{\int }_{{\mathbb{R}}^N}F(x,u)dx,$$

$$⟨J_{\lambda ,a}^{\prime }(u),v⟩$$

Full text

[TABLE]

where $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})$ satisfying $\lim_{|u|\rightarrow\infty}f(x,u)/|u|^{k-1}=q(x)$ uniformly in $x\in\mathbb{R}^{N}$ for any $2<k<2^{\ast}$($2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.

Key words: Nontrivial solutions, Kirchhoff type equations, Caffarelli-Kohn-Nirenberg inequality, Steep potential well.

1 Introduction

Consider the following nonlinear Kirchhoff type equations:

[TABLE]

where $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m$ being positive continuous function on $\mathbb{R}^{+}$, and $f$ is a continuous function on $\mathbb{R}^{N}\times\mathbb{R}$ such that $f(x,s)\equiv 0$ for all $x\in\mathbb{R}^{N}$ and $s<0.$ We assume that the potential $V(x)$ satisfies the following hypotheses:

• $(V1)$

$V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and there exists $c_{0}>0$ such that the set $\{V<c_{0}\}:=\{x\in\mathbb{R}^{N}\ |\ V(x)<c_{0}\}$ has finite positive Lebesgue measure, where $\left|\cdot\right|$ is the Lebesgue measure;

• $(V2)$

$\Omega=intV^{-1}(0)$ is nonempty and has smooth boundary with $\overline{\Omega}=V^{-1}(0).$

Kirchhoff type equations, of the form similar to Eq. $(K_{a,\lambda}),$ are often referred to as being nonlocal because of the presence of the integral. When $m(t)=t,$ Eq. $(K_{a,\lambda})$ is analogous to the stationary case of equations that arise in the study of string or membrane vibrations, namely,

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}.$ As an extension of the classical D’Alembert’s wave equation, Eq. (1) was first presented by Kirchhoff [12] in 1883 to describe the transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations, where $u$ denotes the displacement, $f$ is the external force and $b$ is the initial tension while $a$ is related to the intrinsic properties of the string, such as Young’s modulus.

Since Lions [11] introduced an abstract framework to the Kirchhoff type equations, the qualitative analysis of nontrivial solutions for such equations with various nonlinear terms, including the existence and multiplicity of positive solutions and of sign-changing solution, has begun to receive much attention; see, for example, [1, 2, 4, 13, 14, 15, 16, 17, 20, 21] for the bounded domain case and [5, 7, 8, 9, 10, 18, 19] for the unbounded domain case.

Let us briefly comment some well-known results for the nonlinear Kirchhoff type equations. Alves-Corrêa-Ma [1] studied the following Kirchhoff type equations in the bounded domain $\Omega:$

[TABLE]

They concluded the existence of positive solutions of Eq. (2) when $M$ does not grow too fast in a suitable interval near zero and $f$ is locally Lipschitz subject to some prescribed criteria. Bensedik-Bouchekif [2] studied the asymptotically linear case and obtained the existence of positive solutions of Eq. (2) when the function $M$ is a non-decreasing function and $M(t)\geq m_{0}$ for some $m_{0}>0$, and $f$ is the asymptotically linear satisfying some assumptions about its asymptotic behaviors near zero and infinite. Later, Chen-Kuo-Wu [4] illustrated the difference in the solution behavior which arises from the consideration of the nonlocal effect for Eq. (2) with $M(t)=at+b$ and $f$ being concave-convex nonlinearity.

Compared with the bounded domain case, the unbounded domain case seems to be more delicate. The primary difficulty lies in the lack of the embedding of compactness. Figueiredo-Ikoma-Júnior [7] studied the existence and concentration behaviors of positive solutions to the following Kirchhoff type equations:

[TABLE]

Under suitable conditions on $M$ and $f$, a family of positive solutions for Eq. (3) concentrating at a local minimum of $V$ are constructed. Recently, we [18] introduced the hypotheses $(V1)-(V2)$ to Kirchhoff type equations in $\mathbb{R}^{N}$($N\geq 3$) and studied the existence of nontrivial solution for Eq. $\left(K_{\lambda,a}\right)$ with $m(t)=t$ and $f$ being asymptotically linear or superlinear at infinity on $u.$

Inspired by the above facts, in this paper we are likewise interested in looking for nontrivial solutions for Eq. $(K_{a,\lambda})$ in $\mathbb{R}^{N}$ with $N\geq 1$. However, distinguishing from the existing literatures, we are more focus on the interaction between the functions $m$ and $f,$ leading to the difference in the number of solutions. Specifically, we find that the powers of $m$ and $f$ will dominate the number of solutions for Eq. $(K_{a,\lambda}).$ We require that the function $m$ satisfies some asymptotic behaviors near infinite and that $f$ is $k$-asymptotically linear at infinity on $u$ for any real number $2<k<2^{\ast},$ i.e., $\lim_{|u|\rightarrow\infty}f\left(x,u\right)/|u|^{k-1}=q\left(x\right)$ uniformly in x\in\mathbb{R}^{N},\while not requiring any assumption about the asymptotic behavior near zero of $m$ and $f.$

We wish to point out that in the study of one positive solution, the range of the parameter $a>0$ in Eq. $(K_{a,\lambda})$ is dependent on the limiting function $q$ of $f.$ In other words, the different types of $q$ will bring about the different ranges of $a.$ Moreover, in the study of two positive solutions, the geometry of the variational structure of Eq. $(K_{a,\lambda})$ is known to have a local minimum and a mountain pass, since the power of $m$ is greater than the one of $f.$ In view of this, it is clear to use the minimax method to seek two solutions of Eq. $(K_{a,\lambda})$ as critical points of the associated energy functional $J_{a,\lambda}$. However, since the norms $\|u\|_{D^{1,2}}=\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)^{1/2}$ and $\|u\|_{H^{1}}=\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right)^{1/2}$ are not equivalent in $H^{1}(\mathbb{R}^{N})$, we can not apply the standard techniques to verify the boundedness below of $J_{a,\lambda}$ and the boundedness of the $(PS)$-sequence.

Based on the analysis above, we suggest some new techniques and introduce new hypotheses on $m$ and $q$ in the present paper. By using the minimax method and Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for Eq. $(K_{a,\lambda})$ under the different assumptions on $m$ and $f$, respectively.

We now summarize our main results as follows.

Theorem 1.1

Suppose that $N\geq 1$ and conditions $(V1)-(V2)$ hold. In addition, for any real number $2<k<2^{\ast},$ we assume that the functions $m$ and $f$ satisfy the following conditions:

• $\left(L1\right)$

there exists $m_{\infty}>0$ such that $\lim_{t\rightarrow\infty}t^{-(k-2)/2}m(t)=m_{\infty}$ and

[TABLE]

for all $0\leq\sigma<\eta;$

• $\left(D1\right)$

there exist $q\in L^{\infty}(\mathbb{R}^{N})$ and $0\leq\mu<N-\frac{k(N-2)}{2}$ satisfying $q\not\equiv 0$ on $\overline{\Omega}$ and $\liminf_{\left|x\right|\rightarrow\infty}\left|x\right|^{\mu}q\left(x\right)>0$ such that

[TABLE]

• $\left(D2\right)$

$s\mapsto\frac{f(x,s)}{s^{k-1}}$* is nondecreasing function on $(0,\infty)$ for any fixed $x\in\mathbb{R}.$*

Then there exists $\widetilde{{\Lambda}}>0$ such that Eq. $\left(K_{\lambda,a}\right)$ admits at least one positive solution for all $\lambda>\widetilde{{\Lambda}}$ and $a>0.$

Remark 1.1

$(i)$* It is not difficult to find such functions $m$ satisfying condition $(L1)$. For example, let $m(t)=t^{(k-2)/2}+\theta t^{(k-2)/4}$ for $t>0$, where $2<k<2^{\ast}$ and $\theta\geq 0.$ Clearly, $\lim_{t\rightarrow\infty}t^{-(k-2)/2}m(t)=1$. Moreover, a direct calculation shows that*

[TABLE]

$(ii)$* Under condition $(D1),$ it is not difficult to verify that for any real number $2<k<2^{\ast},$*

[TABLE]

For more details, we refer to the proof of Lemma 3.3 below.

We now assume that the function $m$ satisfies the following assumptions instead of condition $(L1)$:

• $\left(L2\right)$

$m(t)$ is nondecreasing on $t\geq 0;$

• $(L3)$

there exist three positive numbers $m_{0},\delta$ and $T_{0}$ such that $m(t)\geq m_{0}t^{\delta}$ for all $t\geq T_{0}.$

Then we have the following result.

Theorem 1.2

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L2)$ and $(L3)$ with $\delta\geq\frac{2}{N-2}$ hold. In addition, for any real number $2<k<2^{\ast},$ we assume that the function $f$ satisfies conditions $(D1)-(D2).$ Then there exist constants $\widetilde{a}_{\ast},\widetilde{\Lambda}_{\ast}>0$ such that for every $0<a<\widetilde{a}_{\ast}$ and $\lambda>\widetilde{\Lambda}_{\ast},$ Eq. $(K_{a,\lambda})$ admits at least two positive solutions $u_{a,\lambda}^{-}$ and $u_{a,\lambda}^{+}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{-})<0<J_{a,\lambda}(u_{a,\lambda}^{+}).$ In particular, $u_{a,\lambda}^{-}$ is a ground state solution of Eq. $(K_{a,\lambda}).$

Remark 1.2

Compared with Theorem 1.1, if we raise the power $\delta$ of the function $m$ such that $\delta\geq\frac{2}{N-2}>\frac{k-2}{2}$ for $N\geq 3$ in Theorem 1.2, then two positive solutions of Eq. $(K_{a,\lambda})$ can be obtained.

It is well known that for $N\geq 3$ and $2<k<2^{\ast},$ the following minimum problem

[TABLE]

is achieved by some $\overline{\varphi}_{k}\in D^{1,2}(\mathbb{R}^{N})$ by Caffarelli-Kohn-Nirenberg inequality.

We now assume that the following assumption holds:

• $\left(D1\right)^{\prime}$

there exist the function $q(x)$ satisfying $q\not\equiv 0$ on $\overline{\Omega}$ and $q\left(x\right)\leq c_{\ast}|x|^{\frac{k(N-2)}{2}-N}$ for some $c_{\ast}>0$ and for all $x\in\mathbb{R}^{N}$ such that

[TABLE]

Under condition $\left(D1\right)^{\prime}$ and $(\ref{1-1})$, it is easily seen that for any $2<k<2^{\ast},$ the minimum problem

[TABLE]

Then we have the following results.

Theorem 1.3

Suppose that $N\geq 3$ and conditions $(V1)-(V2)$ hold. In addition, for any real number $2<k<2^{\ast},$ we assume that conditions $(L1),(D1)^{\prime}$ and $(D2)$ hold. Then for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}}$ there exists $\overline{\Lambda}>0$ such that Eq. $\left(K_{\lambda,a}\right)$ admits at least one positive solution for all $\lambda>\overline{\Lambda}.$

Theorem 1.4

Suppose that $N\geq 3$ and conditions $(V1)-(V2),(L2)$ hold. In addition, for any real number $2<k<2^{\ast},$ we assume that conditions $(L3)$ with $\delta>\frac{k-2}{2},\left(D1\right)^{\prime}$ and $(D2)$ hold. Then there exists constants $\overline{a}_{\ast},\overline{\Lambda}_{\ast}>0$ such that for every $0<a<\overline{a}_{\ast}$ and $\lambda>\overline{\Lambda}_{\ast},$ Eq. $(K_{a,\lambda})$ admits at least two positive solutions $u_{a,\lambda}^{-}$ and $u_{a,\lambda}^{+}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{-})<0<J_{a,\lambda}(u_{a,\lambda}^{+})$ . In particular, $u_{a,\lambda}^{-}$ is a ground state solution of Eq. $(K_{a,\lambda}).$

Remark 1.3

Unlike Theorem 1.2, in Theorem 1.4 we only require the power $\delta$ of $m(t)$ is greater than $\frac{k-2}{2},$ also leading to two positive solutions of Eq. $(K_{a,\lambda})$ under condition $\left(D1\right)^{\prime}.$

The remainder of this paper is organized as follows. After giving some preliminaries in Section 2, we prove that $J_{\lambda,a}$ satisfies the mountain pass geometry in Section 3. In Sections 4–6, we give the proofs of Theorems 1.1–1.4.

2 Preliminaries

Let

[TABLE]

be equipped with the inner product and norm

[TABLE]

For $\lambda>0,$ we also need the following inner product and norm

[TABLE]

Clearly, $\left\|u\right\|\leq\left\|u\right\|_{\lambda}$ for $\lambda\geq 1.$ Now we set $X_{\lambda}=(X,\left\|u\right\|_{\lambda}).$

For $N=1,2,$ applying conditions $(V1)-(V2)$, the Hölder, Young and Gagliardo-Nirenberg inequalities, there exists a sharp constant $A_{N}>0$ such that

[TABLE]

which shows that

[TABLE]

This implies that

[TABLE]

Similarly, we also have

[TABLE]

For $N\geq 3,$ following [18], we have

[TABLE]

and

[TABLE]

where $S$ is the best constant for the embedding of $D^{1,2}(\mathbb{R}^{N})$ in $L^{2^{\ast}}(\mathbb{R}^{N})$. Set

[TABLE]

Thus, it follows from (6) and (8) that

[TABLE]

which implies that the imbedding $X\hookrightarrow H^{1}(\mathbb{R}^{N})$ is continuous.

Since the imbedding $H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N})$ $(2\leq r<+\infty)$ is continuous for $N=1,2$, from $(\ref{44})$ it follows that for any $r\in[2,+\infty)$,

[TABLE]

for $\lambda\geq\frac{4}{Nc_{0}}\left(1+A_{N}^{8/N}\left|\left\{V<c_{0}\right\}\right|^{2/N}\right)^{-1}$, where $S_{r}$ is the best Sobolev constant for the imbedding of $H^{1}(\mathbb{R}^{N})$ in $L^{r}(\mathbb{R}^{N})$ ($2\leq r<+\infty$). For $N\geq 3,$ following the argument in [18] (see pp 1776-1777), for any $r\in[2,2^{\ast})$ one has

[TABLE]

Now we set

[TABLE]

and

[TABLE]

Thus, by (10)–(11) we have for any $r\in[2,2^{\ast})$ and $\lambda\geq\Lambda_{N},$

[TABLE]

Eq. $(K_{\lambda,a})$ is variational and its solutions are the critical points of the functional defined in $X$ by

[TABLE]

where $\widehat{m}(t)=\int_{0}^{t}m(s)ds$ and $F(x,u)=\int_{0}^{u}f(x,s)ds$. Furthermore, it is not difficult to prove that the functional $J_{\lambda,a}$ is of class $C^{1}$ in $X$, and that

[TABLE]

The following theorem is a variant version of the mountain pass theorem, which helps us find a so-called Cerami type $(PS)$-sequence.

Theorem 2.1

([6], Mountain Pass Theorem) Let $E$ be a real Banach space with its dual space $E^{\ast},$ and suppose that $I\in C^{1}(E,\mathbb{R})$ satisfies

[TABLE]

for some $\mu<\eta,\rho>0$ and $e\in E$ with $\|e\|>\rho.$ Let $\alpha\geq\eta$ be characterized by

[TABLE]

where $\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)=e\}$ is the set of continuous paths joining [math] and $e$. Then there exists a sequence $\{u_{n}\}\subset E$ such that

[TABLE]

3 Mountain pass geometry

In this section we prove that the energy functional $J_{\lambda,a}$ satisfies the mountain pass geometry under the different assumptions on $m$ and $f$, respectively.

Lemma 3.1

Suppose that conditions $(V1)-(V2)$ and $(D1)-(D2)$ hold. Then there exist $\rho>0$ and $\eta>0$ such that \inf\{J_{\lambda,a}(u):u\in X_{\lambda}\with$\ \left\|u\right\|_{\lambda}=\rho\}>\eta$ for all $\lambda\geq\Lambda_{N}.$

Proof. By conditions $(D1)-(D2)$, we obtain that

[TABLE]

and

[TABLE]

Then, by (12) and (16), for every $u\in X$ and $\lambda\geq\Lambda_{N}$ one has

[TABLE]

This implies that

[TABLE]

Thus, letting $\left\|u\right\|_{\lambda}=\rho>0$ small enough, it is easy to obtain that there exists $\eta>0$ such that \inf\{J_{\lambda,a}(u):u\in X_{\lambda}\with$\ \left\|u\right\|_{\lambda}=\rho\}>\eta$ for all $\lambda\geq\Lambda_{N},$ since $2<k<2^{\ast}.$ The lemma is proved.

Lemma 3.2

Suppose that conditions $(V1)-(V2),(D1)^{\prime}$ and $\left(D2\right)$ hold. Then there exist $\rho>0$ and $\eta>0$ such that \inf\{J_{\lambda,a}(u):u\in X_{\lambda}\with$\ \left\|u\right\|_{\lambda}=\rho\}>\eta$ for all $a>0$ and $\lambda\geq\Lambda_{N}.$

Proof. It follows from conditions $\left(D1\right)^{\prime}$ and $\left(D2\right)$ that

[TABLE]

and

[TABLE]

Then, by (4) and (18), for every $u\in X$ and $\lambda\geq\Lambda_{N}$ one has

[TABLE]

which implies that

[TABLE]

Thus, letting $\left\|u\right\|_{\lambda}=\rho>0$ small enough, it is easy to obtain that there exists $\eta>0$ such that \inf\{J_{\lambda,a}(u):u\in X_{\lambda}\with$\ \left\|u\right\|_{\lambda}=\rho\}>\eta$ for all $\lambda\geq\Lambda_{N},$ since $2<k<2^{\ast}.$ The lemma is proved.

Lemma 3.3

Suppose that conditions $(V1)-(V2),(L1)$ and $(D1)-(D2)$ hold. Let $\rho>0$ be as Lemma 3.1. Then there exists $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $a>0$ and $\lambda>0.$

Proof. Let $u\in X\backslash\left\{0\right\}$ with $u>0$ and define $u_{n}\left(x\right)=n^{-N/k}u\left(\frac{x}{n}\right).$ A direct calculation shows that

[TABLE]

and

[TABLE]

Thus, it follows from condition $(D1)$ and Fatou’s lemma that

[TABLE]

which indicates that

[TABLE]

Thus, for each $a>0$, there exists $\phi_{k}\in X\backslash\left\{0\right\}$ with $\phi_{k}>0$ such that

[TABLE]

Using the above inequality, together with conditions $(L1),(D1)-(D2)$ and Lebesgue’s dominated convergence theorem, leads to

[TABLE]

This implies that $J_{\lambda,a}(t\phi_{k})\rightarrow-\infty$ as $t\rightarrow+\infty.$ Therefore, there exists $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ and the lemma is proved.

Note that if condition $(L1)$ is removed, then we can also arrive at a conclusion similar to Lemma 3.3, but the parameter $a>0$ must be small. Now we state this result.

Lemma 3.4

Suppose that conditions $(V1)-(V2)$ and $(D1)-(D2)$ hold. Let $\rho>0$ be as Lemma 3.1. Then there exist $\widetilde{a}_{\ast}>0$ and $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $0<a<\widetilde{a}_{\ast}$ and $\lambda>0.$

Proof. According to the argument of Lemma 3.3, there exists $\phi_{k}\in X\backslash\left\{0\right\}$ with $\phi_{k}>0$ such that $\int_{\mathbb{R}^{N}}q(x)\phi_{k}^{k}dx>0$ by condition $(D1).$ Then using conditions $(D1)-(D2)$, together with Lebesgue’s dominated convergence theorem one has

[TABLE]

where $J_{\lambda,0}(u)=J_{\lambda,a}(u)$ with $a=0.$ Thus, if $J_{\lambda,0}(t\phi_{k})\rightarrow-\infty$ as $t\rightarrow+\infty,$ then there exists $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,0}(e)<0.$ Since $J_{\lambda,a}(e)\rightarrow J_{\lambda,0}(e)$ as $a\rightarrow 0^{+},$ we obtain that there exists $\widetilde{a}_{\ast}>0$ such that $J_{\lambda,a}(e)<0$ for all $0<a<\widetilde{a}_{\ast}$ and $\lambda>0.$

Remark 3.1

We point out that the value of $\widetilde{a}_{\ast}$ can not be determined in general, but only in some special cases. For example, let us assume that $N=4,m\left(t\right)=t$ and

[TABLE]

where the function $q$ is as in condition $\left(D1\right).$ Clearly, the function $m(t)$ does not satisfy condition $(L1).$ Define the minimum problem

[TABLE]

Then $\widehat{\mu}_{1}(\lambda)\geq\frac{\min\left\{1,b\right\}S^{6}}{|q|_{\infty}^{2}\left|\left\{V<c_{0}\right\}\right|^{1/2}}>0$ for all $\lambda>\frac{S^{2}}{c_{0}\left|\left\{V<c_{0}\right\}\right|^{1/2}}.$ Indeed, by $\left(\ref{10}\right),$ for every $\lambda>\frac{S^{2}}{c_{0}\left|\left\{V<c_{0}\right\}\right|^{1/2}}$ there holds

[TABLE]

which implies that $\widehat{\mu}_{1}\geq\frac{\min\left\{1,b\right\}S^{6}}{|q|_{\infty}^{2}\left|\left\{V<c_{0}\right\}\right|^{1/2}}>0.$ Thus, for every $0<a<\frac{2}{9\widehat{\mu}_{1}^{2}(\lambda)}$ there exists $u_{0}\in X_{\lambda}$ such that

[TABLE]

Note that

[TABLE]

where

[TABLE]

A direct calculation shows that $\min_{t>0}g(tu_{0})<0$ by $\left(\ref{3-10}\right).$ This indicates that there exists $t_{0}>0$ such that $J_{\lambda,a}(t_{0}u_{0})<0.$ Letting $e=t_{0}u_{0}\in X_{\lambda}.$ Then $\|e\|_{\lambda}>\rho,$ since $\rho>0$ small enough. Hence, there exsits $e\in X_{\lambda}$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $0<a<\frac{2}{9\widehat{\mu}_{1}^{2}(\lambda)}$ and $\lambda>\frac{S^{2}}{c_{0}\left|\left\{V<c_{0}\right\}\right|^{1/2}}.$

Lemma 3.5

Suppose that conditions $(V1)-(V2),(L1),(D1)^{\prime}$ and $(D2)$ hold. Let $\rho>0$ be as Lemma 3.2. Then for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}},$ there exists $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $\lambda>0.$

Proof. It follows from (5) that for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}},$ there exists $\psi_{k}\in H^{1}(\mathbb{R}^{N})$ with $\psi_{k}>0$ such that

[TABLE]

which implies that

[TABLE]

Using this, together with conditions $(L1),(D1)^{\prime},(D2)$ and Lebesgue’s dominated convergence theorem, yields

[TABLE]

This implies that $J_{\lambda,a}(t\psi_{k})\rightarrow-\infty$ as $t\rightarrow+\infty.$ Hence, for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}},$ there exists $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $\lambda>0$ and the lemma is proved.

If condition $(L1)$ is not required, then we also have a conclusion similar to Lemma 3.5, but $a>0$ must be small.

Lemma 3.6

Suppose that conditions $(V1)-(V2),(D1)^{\prime}$ and $(D2)$ hold. Let $\rho>0$ be as Lemma 3.2. Then there exists $\overline{a}_{\ast}>0$ and $e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $0<a<\overline{a}_{\ast}$ and $\lambda>0.$

Proof. The proof is similar to that of Lemma 3.4, and we omit it here.

Remark 3.2

Similar to Remark 3.1, the value of $\overline{a}_{\ast}$ can also not be determined in general, but only in some special cases. Next, we give an example. For any real number $2<k<2^{\ast},$ we assume that $m\left(t\right)=m_{0}t^{\delta}$ with $\delta>\frac{k-2}{2}$ and

[TABLE]

where the function $q$ satisfies condition $\left(D1\right)^{\prime}.$ It is easily seen that $m\left(t\right)$ does not satisfy condition $(L1).$ Let us consider the minimum problem:

[TABLE]

Then $\widetilde{\mu}_{1}^{(k)}(\lambda)\geq b\left(\overline{\nu}_{1}^{(k)}\right)^{-\frac{2\delta}{2\delta-k+2}}>0$ for all $\lambda>0,$ where $\overline{\nu}_{1}^{(k)}$ is as (4). Indeed, by condition $(D1)^{\prime}$ and the Caffarelli-Kohn-Nirenberg inequality one has

[TABLE]

Using the above inequality leads to for all $\lambda>0,$

[TABLE]

which implies that $\widetilde{\mu}_{1}^{(k)}(\lambda)\geq b\left(\frac{\overline{\nu}_{1}^{(k)}}{c_{\ast}}\right)^{\frac{2\delta}{2\delta-k+2}}>0.$ Thus, for every $0<a<\frac{(\delta+1)(k-2)}{\delta m_{0}k}\left(\frac{2\delta-k+2}{k\delta\widetilde{\mu}_{1}^{(k)}(\lambda)}\right)^{\frac{2\delta-k+2}{k-2}},$ there exists $u_{0}\in X_{\lambda}$ such that

[TABLE]

Next, following the argument in Remark 3.1 we obtain that there exsits $e\in X_{\lambda}$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$ for all $0<a<\frac{(\delta+1)(k-2)}{\delta m_{0}k}\left(\frac{2\delta-k+2}{k\delta\widetilde{\mu}_{1}^{(k)}(\lambda)}\right)^{\frac{2\delta-k+2}{k-2}}$ and $\lambda>0.$

4 Proofs of Theorems 1.1 and 1.2

Define

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Note that $\alpha_{0,a}\left(\Omega\right)$ independent of $\lambda.$ Following the argument in [18], we can take a number $D_{a}>0$ such that $0<\eta\leq\alpha_{\lambda,a}<\alpha_{0,a}(\Omega)<D_{a}$ for all $\lambda\geq\Lambda_{N}$. Thus, by Lemmas 3.1 and 3.3 (or Lemma 3.4) and the mountain pass theorem [6], we obtain that for each $\lambda\geq\Lambda_{N}$ and $a>0$ (or $0<a<\widetilde{{a}}_{\ast}$), there exists a sequence $\left\{u_{n}\right\}\subset X_{\lambda}$ such that

[TABLE]

where $0<\eta\leq\alpha_{\lambda,a}\leq\alpha_{0,a}\left(\Omega\right)<D_{a}.$ Furthermore, we have the following results.

Lemma 4.1

Suppose that conditions $(V1)-(V2),(L1)$ and $(D2)$ hold. Then for each $a>0$ and $\lambda\geq\Lambda_{N},$ the sequence $\{u_{n}\}$ defined in (21) is bounded in $X_{\lambda}.$

Proof. By condition $(D2),$ for $s>0$ one has

[TABLE]

For $n$ large enough, it follows from condition $(L1)$ and $(\ref{3.5})-(\ref{3.11})$ that

[TABLE]

which implies that $\{u_{n}\}$ is bounded in $X_{\lambda}$ for each $a>0$ and $\lambda\geq\Lambda_{N}.$

Lemma 4.2

Suppose that $N\geq 3$, conditions $(V1)-(V2),(L3)$ with $\delta\geq\frac{2}{N-2}$ and $(D1)-(D2)$ hold. Then for all $0<a<\widetilde{{a}}_{\ast}$ and

[TABLE]

the sequence $\{u_{n}\}$ defined in (21) is bounded in $X_{\lambda}.$

Proof. $\left(i\right)$ $\delta=\frac{2}{N-2}:$ Note that $2(\delta+1)=2^{\ast}.$ Suppose on the contrary. Then $\|u_{n}\|_{\lambda}\rightarrow+\infty$ as $n\rightarrow\infty$. The proof is divided into three separate cases:

Case $A:\|u_{n}\|_{D^{1,2}}\rightarrow\infty$ and

[TABLE]

By (21), we have

[TABLE]

where $o(1)$ denotes a quantity which goes to zero as $n\rightarrow\infty.$ Using this, together with condition $(L3)$ and (15), gives

[TABLE]

Note that

[TABLE]

Then there holds

[TABLE]

It follows from (23)-(25) that

[TABLE]

This is a contradiction.

Case $B:\|u_{n}\|_{D^{1,2}}\rightarrow\infty$ and

[TABLE]

Applying (2) and (26)  leads to

[TABLE]

By (24) and (27) one has

[TABLE]

which contradicts with

[TABLE]

Case $C:\int_{\mathbb{R}^{N}}\lambda V(x)u_{n}^{2}dx\rightarrow\infty$ and $\left\|u_{n}\right\|_{D^{1,2}}\leq C_{\ast}$ for some $C_{\ast}>0$ and for all $n.$ From (2), (14) and condition $(L3)$ it follows that

[TABLE]

Applying (2) and the Young inequality gives

[TABLE]

By (29) and the fact of $\|u_{n}\|_{D^{1,2}}\leq C_{\ast}$ for all $n,$ we obtain that

[TABLE]

Using (28) and (30) yields

[TABLE]

which contradicts with

[TABLE]

$\left(ii\right)$ $\delta>\frac{2}{N-2}:$ Clearly, $2(\delta+1)>2^{\ast}.$ Suppose on the contrary. Then $\|u_{n}\|_{\lambda}\rightarrow+\infty$ as $n\rightarrow\infty$. We consider the proof in two separate cases:

Case $D:\|u_{n}\|_{D^{1,2}}\rightarrow\infty.$ It follows from (14)–(15) and condition $(L3)$ that

[TABLE]

By (29), we deduce that

[TABLE]

Using this, together with (31), leads to

[TABLE]

since $2(\delta+1)>2^{\ast}.$ This is a contradiction.

Case $E:\int_{\mathbb{R}^{N}}\lambda V(x)u_{n}^{2}dx\rightarrow\infty$ and $\left\|u_{n}\right\|_{D^{1,2}}\leq C_{\ast}$ for some $C_{\ast}>0$ and for all $n.$ It follows from (14)–(15) and condition $(L3)$ that

[TABLE]

By (30) and (32) one has

[TABLE]

which contradicts with

[TABLE]

In conclusion, the sequence $\{u_{n}\}$ is bounded in $X_{\lambda}$ for all $0<a<\widetilde{{a}}_{\ast}$ and $\lambda>\widetilde{{\Lambda}}_{0}.$ This completes the proof.

We now investigate the following two compactness results for the functional $J_{\lambda,a}$ under conditions $(D1)-(D2)$.

Proposition 4.3

Suppose that $N\geq 1,$ conditions $(V1)-(V2),(L1)$ and $(D1)-(D2)$ hold. Then for each $D>0$ there exists $\widetilde{{\Lambda}}=\widetilde{{\Lambda}}(a,D)\geq\Lambda_{N}>0$ such that $J_{\lambda,a}$ satisfies the $(C)_{\alpha}$–condition in $X_{\lambda}$ for all $\alpha<D$ and $\lambda>\widetilde{{\Lambda}}.$

Proof. Let $\left\{u_{n}\right\}$ be a $\left(C\right)_{\alpha}$-sequence with $\alpha<D.$ By Lemma 4.1, we have $\left\{u_{n}\right\}$ is bounded in $X_{\lambda}.$ Then there exist a subsequence $\left\{u_{n}\right\}$ and $u_{0}\in X_{\lambda}$ such that

[TABLE]

Next, we prove that $u_{n}\rightarrow u_{0}$ strongly in $X_{\lambda}.$ Let $v_{n}=u_{n}-u_{0}.$ Using condition $(V1)$ gives

[TABLE]

Using this, together with the Hölder and Sobolev inequalities, for any $\lambda>\Lambda_{N}$, we check the following estimation:

Case $(i)$ $N=1,2:$

[TABLE]

Case $(ii)$ $N\geq 3:$

[TABLE]

Set

[TABLE]

Clearly, $\Pi_{\lambda,r}\rightarrow 0$ as $\lambda\rightarrow\infty.$ Then we have

[TABLE]

Following the argument of [19], it is easy to verify that

[TABLE]

and

[TABLE]

Thus, using (13), (35) and Brezis-Lieb Lemma [3], we deduce that

[TABLE]

Moreover, it follows from the boundedness of the sequence $\left\{u_{n}\right\}$ in $X_{\lambda}$ that there exists a constant $A>0$ such that $\left\|u_{n}\right\|_{D^{1,2}}^{2}\rightarrow A$ as $n\rightarrow\infty$, which indicates that for any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N}),$ there holds

[TABLE]

as $n\rightarrow\infty$. This implies that

[TABLE]

Note that

[TABLE]

Combining the above two equalities gives

[TABLE]

In addition, it follows from (36), conditions $(L1)$ and $(D3)$ that

[TABLE]

Then there exists a constant $K$ satisfying $K=0$ if $m\left(\left\|u_{0}\right\|_{D^{1,2}}^{2}\right)\geq m\left(A\right)$ or $K<0$ if $m\left(\left\|u_{0}\right\|_{D^{1,2}}^{2}\right)<m\left(A\right)$ such that

[TABLE]

Using this, together with (33), (37), conditions $(L1)$ and $(D3)$ leads to

[TABLE]

which implies that there exists a constant $\widetilde{D}=\widetilde{D}(D)>0$ such that

[TABLE]

It follows from the (15), (34), (37) and (38) that

[TABLE]

which implies that there exists $\widetilde{{\Lambda}}:=\widetilde{{\Lambda}}(a,D)\geq\Lambda_{N}$ such that for $\lambda>\widetilde{{\Lambda}},$

[TABLE]

This completes the proof.

Proposition 4.4

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L2),(L3)$ with $\delta\geq\frac{2}{N-2}$and $(D1)-(D2)$ hold. Then for each $D>0$ there exists $\widetilde{{\Lambda}}_{1}=\widetilde{{\Lambda}}_{1}(a,D)\geq\Lambda_{N}>0$ such that $J_{\lambda,a}$ satisfies the $(C)_{\alpha}$–condition in $X_{\lambda}$ for all $\alpha<D$ and $\lambda>\widetilde{{\Lambda}}_{1}.$

Proof. The proof is similar to that of Proposition 4.3, in which only some places are adjusted. Now we briefly verify it. By Lemma 4.7, there exists a constant $K<0$ such that

[TABLE]

It follows from (33), (37), (39), conditions $(L2)$ and $(D2)$ that

[TABLE]

which implies that for each $\lambda>\Lambda_{N}$ there exists a constant $\overline{D}=\overline{D}(D)>0$ such that

[TABLE]

Next, following the argument of Proposition 4.3, we easily arrive at the conclusion. This completes the proof.

Theorem 4.5

Suppose that $N\geq 1,$ conditions $(V1)-(V2),(L1)$ and $(D1)-(D2)$ hold. Then for every $a>0$ and $\lambda>\widetilde{{\Lambda}},$ the energy functional $J_{a,\lambda}$ has a nontrivial critical point $u_{\lambda}\in X_{\lambda}$ such that $J_{a,\lambda}(u_{\lambda})>0.$

Proof. By Proposition 4.3 and $0<\eta\leq\alpha_{\lambda,a}\leq\alpha_{0,a}\left(\Omega\right)$ for all $\lambda\geq\Lambda_{N}$, $J_{\lambda,a}$ satisfies the $(C)_{\alpha_{\lambda,a}}$–condition in $X_{\lambda}$ for each $a>0$ and $\lambda>\widetilde{{\Lambda}}$. That is, there exist a subsequence $\left\{u_{n}\right\}$ and $u_{\lambda}\in X_{\lambda}$ such that $u_{n}\rightarrow u_{\lambda}$ strongly in $X_{\lambda}.$ This implies that $u_{\lambda}$ is a nontrivial critical point of $J_{\lambda,a}$ satisfying $J_{\lambda,a}\left(u_{\lambda}\right)=\alpha_{\lambda,a}>0.$

Theorem 4.6

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L2),(L3)$ with $\delta\geq\frac{2}{N-2}$ and $(D1)-(D2)$ hold. Then there exists a constant $\widetilde{{\Lambda}}_{2}\geq\max\left\{\widetilde{{\Lambda}}_{0},\widetilde{{\Lambda}}_{1}\right\}$ such that for every $0<a<\widetilde{{a}}_{\ast}$ and $\lambda>\widetilde{{\Lambda}}_{2},$ the energy functional $J_{a,\lambda}$ has a nontrivial critical point $u_{a,\lambda}^{+}\in X_{\lambda}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{+})>0.$

Proof. Similar to the argument of Theorem 4.5, by Proposition 4.4 we easily arrive at the conclusion.

Lemma 4.7

Suppose that $N\geq 3$, conditions $(V1)-{(V2)},(L3)$ with $\delta\geq\frac{2}{N-2}$ and $(D1)-{(D2)}$ hold. Then the energy functional $J_{a,\lambda}$ is bounded below on $X_{\lambda}$ for all $a>0$ and

[TABLE]

Furthermore, if

[TABLE]

then there exists $\widetilde{{R}}_{a}>T_{0}^{1/2}$ such that

[TABLE]

Proof. If $\left\|u\right\|_{D^{1,2}}<T_{0}^{1/2},$ then by (2), (15) and the Young inequality one has

[TABLE]

which shows that $J_{a,\lambda}$ is bounded below on $X_{\lambda}$ for all $a>0$ and $\lambda>\Lambda_{N}.$

If $\left\|u\right\|_{D^{1,2}}\geq T_{0}^{1/2},$ then we consider two cases as follows:

$\left(i\right)$ $\delta=\frac{2}{N-2}$ and $\left\|u\right\|_{D^{1,2}}\geq T_{0}^{1/2}:$ The argument is also divided into two seperate cases:

Case $A:\int_{\mathbb{R}^{N}}\lambda V(x)u^{2}dx\geq\lambda c_{0}S^{-2^{\ast}}\left(\frac{2|q|_{\infty}}{k\lambda c_{0}}\right)^{\frac{2^{\ast}-2}{k-2}}\left\|u\right\|_{D^{1,2}}^{2^{\ast}}.$ By condition $(L3)$, (2), (13) and (16), for each $\lambda>0$ one has

[TABLE]

where we have used the Young and Sobolev inequalities. This implies that $J_{a,\lambda}\left(u\right)$ is bounded below on $X$ for all $a>0$ and $\lambda>\Lambda_{N}.$

Case $B:\int_{\mathbb{R}^{N}}\lambda V(x)u^{2}dx<\lambda c_{0}S^{-2^{\ast}}\left(\frac{2|q|_{\infty}}{k\lambda c_{0}}\right)^{\frac{2^{\ast}-2}{k-2}}\left\|u\right\|_{D^{1,2}}^{2^{\ast}}.$ It follows from (2) that

[TABLE]

Using this, together with condition $(L3)$ once again, yields

[TABLE]

This shows that if

[TABLE]

then $J_{a,\lambda}$ is bounded below on $X_{\lambda}$ for all $a>0$ and there exists $R_{a}>0$ such that

[TABLE]

$\left(ii\right)$ $\delta>\frac{2}{N-2}$ and $\left\|u\right\|_{D^{1,2}}\geq T_{0}^{1/2}:$ It follows from condition $(L3)$, (2), (15) and the Young inequality that

[TABLE]

which implies that $J_{a,\lambda}\left(u\right)$ is bounded below on $X$ for all $a>0$ and $\lambda>\Lambda_{N},$ since $\delta>\frac{2}{N-2}.$ Moreover, for every $a>0,$ there exists

[TABLE]

such that

[TABLE]

Next, we show that there exists a constant $\widehat{R}_{a}>\overline{R}_{a}$ such that

[TABLE]

Let

[TABLE]

where

[TABLE]

For $u\in X_{\lambda}$ with $\left\|u\right\|_{\lambda}\geq\widetilde{{R}}_{a}.$ If $\left\|u\right\|_{D^{1,2}}\geq\overline{R}_{a},$ then the result holds clearly. If $T_{0}^{1/2}\leq\left\|u\right\|_{D^{1,2}}<\overline{R}_{a},$ then it is enough to indicate that $J_{a,\lambda}(u)\geq 0$ when

[TABLE]

Indeed, by (29) we deduce that

[TABLE]

Hence, we obtain that there exists a constant $\widetilde{{R}}_{a}>0$ defined as (40) such that

[TABLE]

This completes the proof.

Lemma 4.8

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L3)$ with $\delta\geq\frac{2}{N-2}$ and $(D1)-(D2)$ hold. Then for every $a>0$ and $\lambda>\widetilde{{\Lambda}}_{4}$ one has

[TABLE]

Proof. The proof directly follows from Lemmas 3.4 and 4.7.

Theorem 4.9

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L2),(L3)$ with $\delta\geq\frac{2}{N-2}$ and $(D1)-(D2)$ hold. Then there exists a constant $\widetilde{{\Lambda}}_{5}\geq\max\left\{\widetilde{{\Lambda}}_{1},\widetilde{{\Lambda}}_{4}\right\}$ such that for every $a>0$ and $\lambda\geq\widetilde{{\Lambda}}_{5},$ $J_{a,\lambda}$ has a nontrivial critical point $u_{a,\lambda}^{-}\in X_{\lambda}$ such that $J_{a,\lambda}(u_{a,\lambda}^{-})=\widetilde{{\theta}}_{a}<0.$

Proof. By Lemma 4.8 and the Ekeland variational principle, there exists a bounded minimizing sequence $\{u_{n}\}\subset X_{\lambda}$ with $\left\|u_{n}\right\|_{\lambda}<\widetilde{{R}}_{a}$ such that

[TABLE]

According to Proposition 4.4, there exist a subsequence $\left\{u_{n}\right\}$ and $u_{a,\lambda}^{-}\in X_{\lambda}$ such that $u_{n}\rightarrow u_{a,\lambda}^{-}$ strongly in $X_{\lambda}.$ This indicates that $u_{\lambda}$ is a nontrivial critical point of $J_{\lambda,a}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{-})=\widetilde{{\theta}}_{a}<0.$

**We are now ready to prove Theorems 1.1 and 1.2: **Theorems 1.1 directly follows from Theorem 4.5. By using Theorems 4.6 and 4.9, there exists a positive constant $\widetilde{{\Lambda}}_{\ast}\geq\max\left\{\widetilde{{\Lambda}}_{2},\widetilde{{\Lambda}}_{5}\right\}$ such that for every $0<a<\widetilde{{a}}_{\ast}$ and $\lambda\geq\widetilde{{\Lambda}}_{\ast},$ Eq. $(K_{a,\lambda})$ admits two positive solutions $u_{a,\lambda}^{-}$ and $u_{a,\lambda}^{+}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{-})<0<J_{a,\lambda}(u_{a,\lambda}^{+}).$ In particular, $u_{a,\lambda}^{-}$ is a ground state solution of Eq. $(K_{a,\lambda}).$ Hence, we arrive at Theorem 1.2.

5 Proofs of Theorems 1.3 and 1.4

Following the argument at the begining of Section 4, by virtue of Lemmas 3.2, 3.5 (or Lemma 3.6) and the mountain pass theorem [6], we obtain that for each $\lambda\geq\Lambda_{N}$ and $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}}$ (or $0<a<\overline{a}_{\ast}$), there exists a sequence $\left\{u_{n}\right\}\subset X_{\lambda}$ such that

[TABLE]

where $0<\eta\leq\alpha_{\lambda,a}\leq\alpha_{0,a}\left(\Omega\right)<D_{a}.$ Furthermore, we have the following result.

Lemma 5.1

Suppose that $N\geq 3,$ conditions $(V1)-(V2),(L3)$ with $\delta>\frac{k-2}{2},(D1)^{\prime}$ and $(D2)$ hold. Then for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}},$ the sequence $\{u_{n}\}$ defined in (41) is bounded in $X_{\lambda}$ for all $\lambda\geq\Lambda_{N}.$

Proof. Suppose on the contrary. Then $\|u_{n}\|_{\lambda}\rightarrow+\infty$ as $n\rightarrow\infty$. We consider the proof in two separate cases:

Case $A:\|u_{n}\|_{D^{1,2}}\rightarrow\infty.$ It follows from (14), (17), condition $(L3)$ and the Caffarelli-Kohn-Nirenberg inequality that

[TABLE]

since $2(\delta+1)>k.$ This is a contradiction.

Case $B:\int_{\mathbb{R}^{N}}\lambda V(x)u_{n}^{2}dx\rightarrow\infty$ and $\left\|u_{n}\right\|_{D^{1,2}}\leq C_{\ast}$ for some $C_{\ast}>0$ and for all $n.$ It follows from (14), (17) and condition $(L3)$ that

[TABLE]

This is a contradiction. In conclusion, the sequence $\{u_{n}\}$ is bounded in $X_{\lambda}$ for all $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}}$ and $\lambda\geq\Lambda_{N}.$ This completes the proof.

Similar to Propositions 4.3 and 4.4, we likewise obtain the following two compactness lemmas for the functional $J_{\lambda,a}$ under conditions $(D1)^{\prime}$ and $(D2)$.

Proposition 5.2

Suppose that $N\geq 3,$ conditions $(V1)-(V3),(L1),(D1)^{\prime}$ and $(D2)$ hold. Then for each $D>0$ there exists $\overline{\Lambda}=\overline{\Lambda}(a,D)\geq\Lambda_{N}>0$ such that $J_{\lambda,a}$ satisfies the $(C)_{\alpha}$–condition in $X_{\lambda}$ for all $\alpha<D$ and $\lambda>\overline{\Lambda}.$

Proposition 5.3

Suppose that $N\geq 3,$ conditions $(V1)-(V3),(L2),(L3)$ with $\delta>\frac{k-2}{2},(D1)^{\prime}$ and $(D2)$ hold. Then for each $D>0$ there exists $\overline{\Lambda}_{\ast}=\overline{\Lambda}_{\ast}(a,D)\geq\Lambda_{N}$ such that $J_{\lambda,a}$ satisfies the $(C)_{\alpha}$–condition in $X_{\lambda}$ for all $\alpha<D$ and $\lambda>\overline{\Lambda}_{\ast}.$

By Proposition 5.2 and 5.3, we now give the following two existence results.

Theorem 5.4

Suppose that $N\geq 3,$ conditions $(V1)-(V3),(L1),(D1)^{\prime}$ and $(D2)$ hold. Then for each $0<a<\frac{1}{m_{\infty}\overline{\mu}_{1}^{(k)}}$ the energy functional $J_{a,\lambda}$ admits a nontrivial critical point $u_{\lambda}\in X_{\lambda}$ such that $J_{a,\lambda}(u_{\lambda})>0$ for all $\lambda>\overline{\Lambda}.$

Proof. Similar to the proof of Theorem 4.5, it is easily proved by using (41), Lemma 4.1 and Proposition 5.2.

Theorem 5.5

Suppose that $N\geq 3,$ conditions $(V1)-(V3),(L2),(L3)$ with $\delta>\frac{k-2}{2},(D1)^{\prime}$ and $(D2)$ hold. Then for every $0<a<\overline{a}_{\ast}$ and $\lambda>\overline{\Lambda}_{\ast},$ the energy functional $J_{a,\lambda}$ has a nontrivial critical point $u_{a,\lambda}^{+}\in X_{\lambda}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{+})>0.$

Proof. Similar to the argument of Theorem 4.6, we easily arrive at the conclusion by Lemma 5.1 and Proposition 5.3.

Lemma 5.6

Suppose that $N\geq 3$, conditions $(V1)-{(V3)},(L3)$ with $\delta>\frac{k-2}{2}$, ${(D1)}^{\prime}$ and ${(D2)}$ hold. Then the energy functional $J_{a,\lambda}$ is bounded below on $X_{\lambda}$ for all $a>0$ and $\lambda>0$. Furthermore, there exists $\overline{R}_{a}>0$ such that

[TABLE]

Proof. If $\left\|u\right\|_{D^{1,2}}<T_{0}^{1/2},$ then by (13) and (3) one has

[TABLE]

which implies that $J_{a,\lambda}$ is bounded below on $X_{\lambda}$ for all $a>0$ and $\lambda>0.$

If $\left\|u\right\|_{D^{1,2}}\geq T_{0}^{1/2},$ then it follows from condition $(L3)$ with $\delta>\frac{k-2}{2},$ (13) and (3) that

[TABLE]

which implies that $J_{a,\lambda}\left(u\right)$ is bounded below on $X$ for all $a>0$ and $\lambda>0,$ since $\delta>\frac{k}{2}-1.$ Moreover, for every $a>0,$ there exists $R_{a}>t_{\overline{B}}:=\left(\frac{2(\delta+1)c_{\ast}}{k\overline{\nu}_{1}^{(k)}m_{0}a}\right)^{1/(2\delta+2-k)}$ such that

[TABLE]

Next, we show that there exists a constant $\overline{R}_{a}>0$ such that $J_{a,\lambda}(u)\geq 0$ for all $u\in X_{\lambda}$ with $\left\|u\right\|_{\lambda}\geq\overline{R}_{a}.$ Let

[TABLE]

For $u\in X_{\lambda}$ with $\left\|u\right\|_{\lambda}\geq\overline{R}_{a}.$ If $\left\|u\right\|_{D^{1,2}}\geq R_{a},$ then the result holds clearly. If $\left\|u\right\|_{D^{1,2}}<R_{a},$ then it is enough to indicate that $J_{a,\lambda}(u)\geq 0$ when $\int_{\mathbb{R}^{N}}\lambda V(x)u^{2}dx\geq\frac{2c_{\ast}}{k\overline{\nu}_{1}^{(k)}}R_{a}^{k}.$ Indeed, we have

[TABLE]

Hence, we obtain that there exists a constant $\overline{R}_{a}>0$ defined as (42) such that

[TABLE]

This completes the proof.

Lemma 5.7

Suppose that $N\geq 3,$ conditions $(V1)-(V3),(L3)$ with $\delta>\frac{k-2}{2},(D1)^{\prime}$ and $(D2)$ hold. Then for every $a>0$ and $\lambda>0$ one has

[TABLE]

Proof. The proof directly follows from Lemmas 3.6 and 5.6.

Theorem 5.8

Suppose that $N\geq 3$, conditions $(V1)-(V3),(L2),(L3)$ with $\delta>\frac{k-2}{2},(D1)^{\prime}$ and $(D2)$ hold.. Then for every $a>0$ and $\lambda>\overline{\Lambda}_{\ast},$ $J_{a,\lambda}$ has a nonzero critical point $u_{a,\lambda}^{-}\in X_{\lambda}$ such that

[TABLE]

where $\overline{\theta}_{a}$ is as in (43).

Proof. By Lemma 5.7 and the Ekeland variational principle, we obtain that there exists a minimizing bounded sequence $\{u_{n}\}\subset X_{\lambda}$ with $\|u_{n}\|_{\lambda}<\overline{R}_{a}$ such that

[TABLE]

Then from Proposition 5.3 it follows that there exist a subsequence $\{u_{n}\}$ and $u_{a,\lambda}^{-}\in X_{\lambda}$ with $\|u_{a,\lambda}^{-}\|_{\lambda}<\overline{R}_{a}$ such that $u_{n}\rightarrow u_{a,\lambda}^{-}$ strongly in $X_{\lambda}.$ This indicates that $J_{a,\lambda}^{\prime}(u_{a,\lambda}^{-})=0$ and $J_{a,\lambda}(u_{a,\lambda}^{-})=\overline{\theta}_{a}<0.$ The proof is complete.

**We are now ready to prove Theorems 1.3 and 1.4: **Theorems 1.3 directly follows from Theorem 5.4. By virtue of Theorems 5.5 and 5.8, for every $0<a<\overline{a}_{\ast}$ and $\lambda>\overline{\Lambda}_{\ast},$ Eq. $(K_{a,\lambda})$ admits two positive solutions $u_{a,\lambda}^{-}$ and $u_{a,\lambda}^{+}$ satisfying $J_{a,\lambda}(u_{a,\lambda}^{-})<0<J_{a,\lambda}(u_{a,\lambda}^{+}).$ In particular, $u_{a,\lambda}^{-}$ is a ground state solution of Eq. $(K_{a,\lambda}).$ Hence, Theorem 1.4 is proved.