On the Kirchhoff type equations in $\mathbb{R}^{N}$

TL;DR

This paper investigates the existence, multiplicity, and nonexistence of positive solutions for a nonlinear Kirchhoff type equation in \\mathbb{R}^N, emphasizing the geometric properties of the associated energy functional and how solutions depend on parameters and dimension.

Contribution

It provides new insights into the geometric analysis of the energy functional and establishes conditions for positive solutions based on the dimension and parameters, extending previous results.

Findings

Unique positive solution for 1 ≤ N ≤ 4.

At least two positive solutions for N ≥ 5.

Dependence of solution existence on parameters a and dimension N.

Abstract

Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) \Delta u+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2<p<\min \left\{ 4,2^{\ast }\right\}$($2^{\ast }=\infty $ for $N=1,2$ and $2^{\ast }=2N/(N-2)$ for $N\geq 3)$ and the function $f\in C(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1\leq N\leq4$ while at least two positive solutions are permitted for $N\geq5$.

Equations

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

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

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

$$\underset{|u|\to \infty }{\lim }\frac{{\int }_0^{u}h(x,s)ds}{u^4}=\infty \text{ uniformly in }x\in {\mathbb{R}}^N,$$

$$\exists \mu >4\text{ such that }0<\mu {\int }_0^{u}h(x,s)ds\le h(x,u)u\text{ for }u\ne 0.$$

$$\overline{\mathbf{M}}=\{u\in H_{rad}^1({\mathbb{R}}^3)\backslash \{0\}:2⟨I^{\prime }(u),u⟩-P(u)=0\},$$

$$a\le {\left(\frac{N-4}{N-2}\right)}^{\frac{N-2}{2}}\frac{2}{(N-4)b^{\frac{N-4}{2}}{\int }_{{\mathbb{R}}^N}|\nabla \overline{u}{|}^{2}dx}.$$

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

$$J_a\left(u\right)=\frac{a}{4}{\left({\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx\right)}^2+\frac{1}{2}{\int }_{{\mathbb{R}}^N}(b|\nabla u{|}^2+u^2)dx-\frac{1}{p}{\int }_{{\mathbb{R}}^N}f(x)|u{|}^{p}dx.$$

$$⟨J_a^{\prime }(u),\phi ⟩=\left(a{\int }_{{\mathbb{R}}^N}|\nabla u{|}^{2}dx+b\right){\int }_{{\mathbb{R}}^N}\nabla u\nabla \phi dx+{\int }_{{\mathbb{R}}^N}(u\phi -f(x)|u{|}^{p-2}u\phi )dx$$

$${\overline{A}}_f(u)=\frac{{\left({\int }_{{\mathbb{R}}^N}f(x)|u{|}^{p}dx\right)}^{2/(p-2)}}{{\Vert u\Vert }_{D^{1,2}}^4{\Vert u\Vert }_{H^1}^{2(4-p)/(p-2)}}\text{ for }N\ge 4$$

$${\underline{A}}_f(u)=\frac{{\left({\int }_{{\mathbb{R}}^N}{f}_{\min }|u{|}^{p}dx\right)}^{2/(p-2)}}{{\Vert u\Vert }_{D^{1,2}}^4{\Vert u\Vert }_{H^1}^{2(4-p)/(p-2)}}\text{ for }N=4,$$

$${\overline{\mathbf{A}}}_f=\underset{u\in H^1({\mathbb{R}}^N)\backslash \left\{0\right\}}{\sup }{\overline{A}}_f(u)\text{ and }{\underline{\mathbf{A}}}_f=\underset{u\in H^1({\mathbb{R}}^N)\backslash \left\{0\right\}}{\sup }{\underline{A}}_f(u).$$

$${\overline{A}}_f(u)$$

$${\underline{A}}_f\left(u\right)\le {\left(f_{\min }{C}_p^p\right)}^{\frac{2}{p-2}}{\left(\frac{{\Vert u\Vert }_{L^2}^2}{{\Vert u\Vert }_{D^{1,2}}^2+{\Vert u\Vert }_{L^2}^2}\right)}^{\frac{4-p}{p-2}}\le {\left(f_{\min }{C}_p^p\right)}^{\frac{2}{p-2}},$$

$$0<{\overline{\mathbf{A}}}_f\le {\overline{C}}_0(N,p,f)\text{ for }N\ge 4$$

$$0<{\underline{\mathbf{A}}}_f\le {\underline{C}}_0(p,f)\text{ for }N=4.$$

$${\overline{a}}_{\ast }=\frac{2(p-2)}{4-p}{\left(\frac{4-p}{p}\right)}^{2/(p-2)}{\overline{\mathbf{A}}}_f\text{ for }N\ge 4$$

$${\underline{a}}_{\ast }=\frac{2(p-2)}{4-p}{\left(\frac{4-p}{p}\right)}^{2/(p-2)}{\underline{\mathbf{A}}}_f\text{ for }N=4.$$

$$a>0$$

$$J_a^{\infty }(v_a^{+})=\underset{u\in H^1({\mathbb{R}}^N)\backslash \{0\}}{\inf }{J}_a^{\infty }(u)<0,$$

$$J_a(u_a^{+})=\underset{u\in H^1({\mathbb{R}}^N)\backslash \{0\}}{\inf }{J}_a(u)<0.$$

$${\mathbf{M}}_a(c)=\{u\in {\mathbf{M}}_a:J_a(u)<c\}\text{ for some }c>0,$$

$${\mathbf{M}}_a^{(1)}(c)=\{u\in {\mathbf{M}}_a(c):\Vert u{\Vert }_{H^1}<C_1\}\text{and}{\mathbf{M}}_a^{(2)}(c)=\{u\in {\mathbf{M}}_a(c):\Vert u{\Vert }_{H^1}>C_2\},$$

$$D(p)=\left\{\begin{array}[]{ll}\left(\frac{4-p}{2}\right)^{1/(p-2)},&\text{ if }2<p\leq 3,\\ \frac{1}{2},&\text{ if }3<p<\min\{4,2^{\ast}\}.\end{array}\right.$$

$$\frac{1}{2}\le D(p)<\frac{1}{\sqrt{e}}\text{ and }D(p){\left(\frac{2}{4-p}\right)}^{2/(p-2)}>1.$$

$${\Lambda }_0=\left[1-D(p){\left(\frac{f_{\max }}{f_{\infty }}\right)}^{2/(p-2)}\right]{\left(\frac{f_{\infty }}{S_p^p}\right)}^{2/(p-2)},$$

$${\Lambda }_0=(1-D(p)){\left(\frac{f_{\infty }}{S_p^p}\right)}^{2/(p-2)}.$$

$$\Lambda=\left\{\begin{array}[]{ll}\frac{4-p}{2}\left(\frac{f_{\infty}(4-p)}{2pS_{p}^{p}}\right)^{2/(p-2)}&\text{ if }N=1,2,3,\\ \min\left\{\frac{p-2}{2(4-p)}\left(\frac{4-p}{p}\right)^{2/(p-2)}\Lambda_{0},\overline{a}_{\ast}\right\}&\text{ if }N\geq 4.\end{array}\right.$$

$$\begin{array}[]{ll}-\Delta u+u=f_{\infty}|u|^{p-2}u&\text{ in }\mathbb{R}^{N}.\end{array}$$

Full text

[TABLE]

where $N\geq 1,a,b>0,2<p<\min\left\{4,2^{\ast}\right\}$($2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3)$ and the function $f\in C(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1\leq N\leq 4$ while at least two positive solutions are permitted for $N\geq 5$.

1 Introduction

We are concerned with the following nonlinear Kirchhoff type equations:

[TABLE]

where $N\geq 1,a,b>0,$ $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $h\in C(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}).$

Kirchhoff type equations, of the form similar to Eq. $(\ref{1-0}),$ are 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. $(\ref{1-1})$ was first presented by Kirchhoff [17] 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, $h$ 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. Equations of this type are often referred to as being nonlocal because of the presence of the integral.

After the pioneering work of Pohozaev [27] and Lions [21], the solvability of the Kirchhoff type equation $(\ref{1-1})$ has been well-studied in general dimension by various authors, see for examples, D’Ancona-Shibata [7], D’Ancona-Spagnolo [8] and Nishihara [26]. More recently, the corresponding elliptic version like Eq. $(\ref{1-0})$ has begun to receive much attention via variational methods. We refer the reader to [1, 2, 5, 10, 12, 14, 15, 18, 19, 20, 25, 29, 30, 31, 34, 36] and the references therein.

Most of researchers have of late years focused on the existence of positive solutions, ground states, radial solutions and semiclassical states for Eq. $(\ref{1-0})$ in lower dimensions, i.e., $N=1,2,3.$ The typical way to deal with such problem is to apply the mountain-pass theorem or the Nehari manifold method. Owing to the fourth power of the nonlocal term, one usually assumes that the nonlinearity $h(x,u)$ is either $4$-superlinear at infinity on $u$ in the sense that

[TABLE]

or satisfies the following (AR)-condition:

[TABLE]

For example, $h(x,u)=f(x)\left|u\right|^{p-2}u$ with $4<p<2^{\ast}(2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3).$ By so doing, one can easily verify the mountain-pass geometry and the boundedness of (PS) sequences for the energy functional. However, there have a large number of functions $h(x,u)$ not satisfying the above assumptions, such as $h(x,u)=f(x)\left|u\right|^{p-2}u(2<p<\min\left\{4,2^{\ast}\right\}).$ For that reason, some other approaches need to be introduced in this case.

In studying the radial solutions for a class of autonomous Schrödinger-Poisson systems in $\mathbb{R}^{3},$ Ruiz [28] established a manifold as follows

[TABLE]

where $I$ is the energy functional and $P(u)$ is the Pohozaev identity corresponding to the system. It is usually called the Nehari-Pohozaev manifold which is different from the Nehari manifold. By restricting the energy functional to such manifold, the boundedness of (PS) sequences can be solved effectively when the nonlinearity does not satisfy the (AR)-condition above.

Inspired by Ruiz [28], Li-Ye [19] applied the Nehari-Pohozaev manifold (slightly different from $\overline{\mathbf{M}}$) to Kirchhoff type equations in $\mathbb{R}^{3}$. By using the constraint minimization method, together with the monotonicity trick by Jeanjean [16], they found one ground state solution with positive energy of Eq. $(\ref{1-0})$ when $V(x)\leq\liminf_{|y|\rightarrow\infty}V(y)=V_{\infty}<\infty$ and $h(x,u)=\left|u\right|^{p-2}u(3<p<2^{\ast})$. Later, using the similar approach to that in [19], Guo [12] and Tang-Chen [34] also obtained the existence of ground state solutions for Eq. $(\ref{1-0})$ with a general nonlinearity $h(x,u)\equiv h(u)$ in $\mathbb{R}^{3}$, respectively. In addition, Ye [36] proved the existence of high energy solutions for Eq. $(\ref{1-0})$ with $h(x,u)\equiv h(u)$ in $\mathbb{R}^{3}$ via the Nehari-Pohozaev manifold and the linking theorem. It is worthy noting that the nonlinearity $h(u)$ given by [12, 34, 36] can cover the power functions $\left|u\right|^{p-2}u(2<p<4).$

Azzollini [1, 2] investigated a class of autonomous Kirchhoff type equations in higher dimensions $N\geq 3,$ i.e., Eq. $(\ref{1-0})$ with $V(x)\equiv 0$ and $h(x,u)\equiv h(u)$ satisfying the Berestycki–Lions type conditions (see [3]). With the aid of the radial ground state solution $\overline{u}$ to the semilinear elliptic equation $-\Delta u=h(u)$ in $\mathbb{R}^{N}(N\geq 3)$, the following results were obtained:

$(i)$ $N=3:$ one radial ground state solution exists for all $a,b>0;$

$(ii)$ $N=4:$ one radial ground state solution exists if and only if $a<\left(\int_{\mathbb{R}^{N}}|\nabla\overline{u}|^{2}dx\right)^{-1};$

$(iii)$ $N\geq 5:$ one radial solution exists if and only if

[TABLE]

Motivated by these findings mentioned above, in the present paper we are likewise interested in looking for positive solutions of Kirchhoff type equations. The problem we consider is thus

[TABLE]

where $N\geq 1,a,b>0,2<p<\min\left\{4,2^{\ast}\right\}$ and the function $f(x)$ satisfies:

• $(D1)$

$f\in C(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ with $f_{\min}=\inf_{x\in\mathbb{R}^{N}}f\left(x\right)>0.$

Eq. $(E_{a})$ is variational, and its solutions correspond to critical points of the energy functional $J_{a}:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ given by

[TABLE]

Furthermore, one can see that $J_{a}$ is a $C^{1}$ functional with the derivative given by

[TABLE]

for all $\varphi\in H^{1}(\mathbb{R}^{N})$, where $J_{a}^{\prime}$ denotes the Fréchet derivative of $J_{a}.$

Distinguishing from the existing literature, this paper is devoted to study a series of questions as follows:

• $(I)$

In spite of the amount of papers dealing with Eq. $(\ref{1-0})$, the geometric properties of the energy functional $J_{a}$ have not been described in detail. One objective of this study is to shed some light on the behavior of $J_{a}$. We will study whether $J_{a}$ is bounded below or not, depending on the parameter $a$ and the dimension $N.$

• $(II)$

As we can see, the Nehari-Pohozaev manifold can help to find positive solutions with positive energy for Eq. $(\ref{1-0})$ when the nonlinearity $h(x,u)$ does not satisfy the (AR)-condition (1.3) (see [12, 19, 34, 36]). However, to our knowledge, such approach is only valid for the case of $N=3$. In our study, since the nonlinearity $f(x)\left|u\right|^{p-2}u(2<p<\min\left\{4,2^{\ast}\right\})$ does not satisfy the (AR)-condition (1.3) as well, we would like to know whether there exists an approach to study the existence of positive solution with positive energy of Eq. $(E_{a})$ in any dimensions $N\geq 1.$

• $(III)$

According to the geometry of the energy functional $J_{a},$ we think that $J_{a}$ should have two critical points in some dimensions $N,$ where one is a global minimizer with negative energy and the other one is a local minimizer with positive energy. In view of this, another objective of this study is to explore the existence of two positive solutions for Eq. $(E_{a})$, which seems not to be involved in the literature.

In what follows, without loss of generality, we always assume $b=1.$ For any $u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\},$ we define

[TABLE]

and

[TABLE]

where $\left\|u\right\|_{H^{1}}=\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right)^{1/2}$ and $\left\|u\right\|_{D^{1,2}}=\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)^{1/2}.$ Since $\overline{A}_{f}$ and $\underline{A}_{f}$ are [math]-homogeneous, we can denote the extremal values by

[TABLE]

Applying the Gagliardo-Nirenberg and Young inequalities gives

[TABLE]

and

[TABLE]

where $f_{\max}:=\sup_{x\in\mathbb{R}^{N}}f\left(x\right),\left\|u\right\|_{L^{2}}=\left(\int_{\mathbb{R}^{N}}u^{2}dx\right)^{1/2}$and $C_{p}>0$ is a sharp constant of Gagliardo-Nirenberg inequality. Thus, there exist two positive numbers $\overline{C}_{0}(N,p,f)$ and $\underline{C}_{0}(p,f)$ such that

[TABLE]

and

[TABLE]

Let

[TABLE]

and

[TABLE]

We now summarize the first part of our main results as follows.

Theorem 1.1

*Suppose that $2<p<\min\left\{4,2^{\ast}\right\}$ and condition $(D1)$ holds. Then the following statements are true.

$(i)$ If $N=1,2,3,$ then $J_{a}$ is not bounded below on $H^{1}(\mathbb{R}^{N})$ for all $a>0;$

$(ii)$ If $N=4,$ then for each $0<a<\underline{a}_{\ast},$ $J_{a}$ is not bounded below on $H^{1}(\mathbb{R}^{N}),$ whereas for each $a>\overline{a}_{\ast},$ $J_{a}$ is bounded below on $H^{1}(\mathbb{R}^{N})$ and $\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)>0;$

$(iii)$ If $N\geq 5,$ then $J_{a}$ is bounded below on $H^{1}(\mathbb{R}^{N})$ for all $a>0.$ More precisely, for each $0<a<\overline{a}_{\ast},$ there holds $-\infty<\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)<0,$ whereas for each $a>\overline{a}_{\ast},$ there holds $\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)>0.$*

For brevity, we sum up the main result of Theorem 1.1 with the table below:

[TABLE]

Note that $\frac{p^{2/\left(p-2\right)}}{2^{p/\left(p-2\right)}}>1,$ since $2<p<2^{\ast}$ for $N\geq 4.$ From Theorem 1.1 $(ii)-(iii)$, one can see that $\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)>0$ for $a>\overline{a}_{\ast}$, if $N\geq 4$. However, we obtain the following nonexistence result.

Theorem 1.2

Suppose that $N\geq 4$ and condition $(D1)$ holds. Then for each $a>\frac{p^{2/\left(p-2\right)}}{2^{p/\left(p-2\right)}}\overline{a}_{\ast},$ Eq. $(E_{a})$ does not admit any nontrivial solutions.

Next, we need the following assumption on $f.$

• $\left(D2\right)$

\lim_{\left|x\right|\rightarrow\infty}f\left(x\right)=f_{\infty}>0\uniformly on$\ \mathbb{R}^{N}.$

Theorem 1.3

$(i)$* Suppose that $N\geq 5$ and $f(x)\equiv f_{\infty}>0.$ Then for each $0<a<\overline{a}_{\ast},$ Eq. $(E_{a})$ has a positive ground state solution $v_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ satisfying*

[TABLE]

*where $J_{a}^{\infty}=J_{a}$ with $f(x)\equiv f_{\infty}.$

$(ii)$ Suppose that $N\geq 5$ and conditions $(D1)-(D2)$ hold. In addition, we assume that*

• $(D3)$

$\int_{\mathbb{R}^{N}}(f(x)-f_{\infty})(v_{a}^{+})^{p}dx>0,$* where $v_{a}^{+}$ is the positive solution as described in part $(i)$.*

Then for each $0<a<\overline{a}_{\ast},$ Eq. $(E_{a})$ has a positive ground state solution $u_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

In order to obtain the existence of positive solution with positive energy for Eq. $(E_{a}),$ it is necessary to introduce the filtration of the Nehari manifold. That is,

[TABLE]

where $\mathbf{M}_{a}=\{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}:\left\langle J_{a}^{\prime}(u),u\right\rangle=0\}$ is the Nehari manifold. We will show that $\mathbf{M}_{a}(c)$ can be divided into two parts

[TABLE]

in which each local minimizer of the functional $J_{a}$ is a critical point of $J_{a}$ in $H^{1}(\mathbb{R}^{N})$. Our approach is to minimize the energy functional $J_{a}$ on $\mathbf{M}_{a}^{(1)}(c)$, where $J_{a}$ is bounded below and the minimizing sequence is bounded. In fact, such approach has been applied in the study of Schrodinger-Poisson systems in $\mathbb{R}^{3}$ by us (see [32, 33]).

We assume that $f$ satisfies the following condition:

• $\left(D4\right)$

$f_{\max}=\sup_{x\in\mathbb{R}^{N}}f\left(x\right)<\frac{f_{\infty}}{D(p)^{(p-2)/2}},$ where

[TABLE]

Remark 1.4

By a direct calculation, we obtain that for $2<p<4,$

[TABLE]

Let

[TABLE]

where $S_{p}$ is the best Sobolev constant for the embedding of $H^{1}(\mathbb{R}^{N})$ in $L^{p}(\mathbb{R}^{N}).$ In particular, if $f(x)\equiv f_{\infty},$ then equality (1.6) becomes

[TABLE]

Set

[TABLE]

Let $w_{0}$ be the unique positive solution of the following Schrödinger equation

[TABLE]

From [17], we see that

[TABLE]

and

[TABLE]

where $J_{0}^{\infty}$ is the energy functional of equation $(E_{0}^{\infty})$ in $H^{1}(\mathbb{R}^{N})$ in the form

[TABLE]

We now summarize the second part of our main results as follows.

Theorem 1.5

*Assume that $f(x)\equiv f_{\infty}>0.$ Then the following statements are true.

$\left(i\right)$ If $N\geq 1,$ then for each $0<a<\Lambda,$ Eq. $(E_{a})$ has at least a positive solution $v_{a}^{-}\in H^{1}(\mathbb{R}^{N})$ satisfying*

[TABLE]

$\left(ii\right)$* If $1\leq N\leq 4,$ then for each $0<a<\Lambda,$ Eq. $(E_{a})$ has a unique positive solution $v_{a}^{-}\in H^{1}(\mathbb{R}^{N}),$ which is radially symmetric;

$\left(iii\right)$ If $N\geq 5,$ then for each $0<a<\Lambda,$ Eq. $(E_{a})$ has at least two positive solutions $v_{a}^{-},v_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ satisfying*

[TABLE]

and $J_{a}^{\infty}(v_{a}^{+})<0<J_{a}^{\infty}(v_{a}^{-}).$ In particular, $v_{a}^{+}$ is a ground state solution of Eq. $(E_{a}).$

With the aid of Theorem 1.5, we obtain the following results in the nonautnomous case.

Theorem 1.6

Suppose that $N\geq 1$ and conditions $(D1)-(D2),(D4)$ hold. In addition, we assume that

• $(D5)$

$\int_{\mathbb{R}^{N}}(f(x)-f_{\infty})(v_{a}^{-})^{p}dx>0,$* where $v_{a}^{-}$ is the positive solution as described in Theorem 1.5.*

Then for each $0<a<\Lambda,$ Eq. $(E_{a})$ has at least a positive solution $u_{a}^{-}\in H^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Theorem 1.7

Suppose that $N\geq 5$ and conditions $(D1)-(D5)$ hold. Then for each $0<a<\Lambda,$ Eq. $(E_{a})$ has at least two positive solutions $u_{a}^{-},u_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

and

[TABLE]

In particular, $u_{a}^{+}$ is a ground state solution of Eq. $(E_{a}).$

Theorem 1.8

*Let $u_{a}^{-}$ be the positive solution of Eq. $(E_{a})$ as described in Theorem 1.5 or 1.6. Then we have the following conclusions:

$\left(i\right)$ If $N=1$ and $f(x)$ is weakly differentiable satisfying*

[TABLE]

*then $u_{a}^{-}$ is a positive ground state solution of Eq. $(E_{a}).$

$\left(ii\right)$ If $N=2$ and $f(x)$ is weakly differentiable satisfying*

[TABLE]

then $u_{a}^{-}$ is a positive ground state solution of Eq. $(E_{a}).$

The following table sums up the above main results:

[TABLE]

In the above table, ”Two solutions” (respectively ”One solution”) means that there exist at least two (respectively one) positive solutions. On the other hand, ”No solution” means that there are no nontrivial solutions.

Recently, Azzollini [1, 2] has proved that Eq. $(E_{a})$ with $f(x)\equiv f_{\infty}$ admits a ground state solution with positive energy for all $a>0$ when $N=3$ and for $a>0$ sufficiently small when $N=4.$ In the following, we shall further describe some characteristics of such solution depending on $a$ and $f_{\infty}$, which are not concerned in [1, 2].

Define the fibering map $h_{a,u}:t\rightarrow J_{a}\left(tu\right)$ as

[TABLE]

About its theory and application, we refer the reader to [4, 9]. Note that for $u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\}$ and $t>0,$ $h_{a,u}^{\prime}\left(t\right)=0$ holds if and only if $tu\in\mathbf{M}_{a}$. In particular, $h_{a,u}^{\prime}\left(1\right)=0$ holds if and only if $u\in\mathbf{M}_{a}.$ It is natural to split $\mathbf{M}_{a}$ into three parts corresponding to the local minima, local maxima and points of inflection. As a consequence, following [35], we can define

[TABLE]

For $2<p<4,$ we set

[TABLE]

It is clearly that $\overline{A}_{0}>\Lambda.$ We now state the last part of our main results as follows.

Theorem 1.9

*Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$ with $f(x)\equiv f_{\infty}$. Then the following statements are true.

$(i)$ When $N=3,$ for each $a>0$ with $\sqrt{a^{2}+4}+\frac{2}{a}\geq A_{0},$ there holds $u_{0}\in\mathbf{M}_{a}^{-}.$ In particular, $v_{a}^{-}$ is a ground state solution as in Theorem 1.5 $\left(i\right)$ for $N=3.$

$(ii)$ When $N=4,$ for each $0<a\leq\overline{A}_{0},$ there holds $u_{0}\in\mathbf{M}_{a}^{-},$ whereas for each $a>A_{0}^{\ast},$ there holds $u_{0}\in\mathbf{M}_{a}^{+}.$ In particular, $v_{a}^{-}$ is a ground state solution as in Theorem 1.5 $\left(i\right)$ for $N=4.$*

Remark 1.10

$(i)$* Note that $\inf_{a>0}\left(\sqrt{a^{2}+4}+\frac{2}{a}\right)>0.$ Then when*

[TABLE]

*there holds $\sqrt{a^{2}+4}+\frac{2}{a}\geq A_{0}$ for all $a>0.$ This shows that $u_{0}\in\mathbf{M}_{a}^{-}$ for all $a>0.$

$(ii)$ If there is a number $a_{0}>0$ such that $\sqrt{a_{0}^{2}+4}+\frac{2}{a_{0}}<A_{0},$ then there exist positive numbers $a_{1},a_{2}$ such that $\sqrt{a^{2}+4}+\frac{2}{a}\geq A_{0}$ for all $a\in(0,a_{1}]\cup[a_{2},\infty)$ and $\sqrt{a^{2}+4}+\frac{2}{a}<A_{0}$ for all $a\in(a_{1},a_{2}).$*

The structure of this paper is as follows. After briefly introducing some technical lemmas in Section 2, we prove Theorems 1.1–1.3 in Section 3, and demonstrate proofs of Theorems 1.5 and 1.6 in Sections 4 and 5, respectively. Section 6 is dedicated to the proof of Theorems 1.8 and 1.9.

2 Preliminaries

Firstly, we consider the boundedness below for energy functional $J_{a}$ on $H^{1}(\mathbb{R}^{N})$ for $N\geq 4.$ For $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\},$ we define

[TABLE]

Lemma 2.1

*Suppose that $N\geq 4$ and condition $(D1)$ holds. Then the following results are true.

$\left(i\right)$ For each $0<a<\overline{a}_{\ast}$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\},$ there exists a constant $\widehat{t}_{a}^{(0)}>\left(\frac{p}{4-p}\right)^{1/(p-2)}T_{f}(u)$ such that*

[TABLE]

$\left(ii\right)$* For each $a\geq\overline{a}_{\ast}$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\},$ there holds $J_{a}\left(tu\right)\geq 0$ for all $t>0.$*

Proof. $\left(i\right)$ For $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}$ and $t>0$, it has

[TABLE]

where

[TABLE]

Clearly, $J_{a}(tu)=0$ if and only if

[TABLE]

It is easy to see that

[TABLE]

where $\hat{t}_{a}=\left(\frac{p}{2}\right)^{1/(p-2)}T_{f}(u).$ By calculating the derivative of $g(t)$, we obtain

[TABLE]

which implies that $g(t)$ is decreasing when $0<t<\left(\frac{p}{4-p}\right)^{1/(p-2)}T_{f}(u)$ and is increasing when $t>\left(\frac{p}{4-p}\right)^{1/(p-2)}T_{f}(u).$ This indicates that

[TABLE]

Note that

[TABLE]

Then there exists $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}$ such that

[TABLE]

Using the above inequality, together with $(\ref{3-5}),$ leads to

[TABLE]

This implies that there exist two numbers $\widehat{t}_{a}^{(0)}$ and $\widehat{t}_{a}^{(1)}$ satisfying

[TABLE]

such that

[TABLE]

That is,

[TABLE]

Thus,

[TABLE]

and so

[TABLE]

Note that

[TABLE]

Then we have

[TABLE]

and

[TABLE]

Consequently, we arrive at inequality (2.2).

$\left(ii\right)$ For each $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\},$ we can find a unique $t_{0}:=t_{0}(u)>0$ such that $h_{a,u}(t_{0})=0$ and $h_{a,u}^{\prime}(t_{0})=0.$ In fact, we only need to solve the system with respect to the variables $t,a$

[TABLE]

A direct calculation shows that

[TABLE]

and accordingly,

[TABLE]

Since

[TABLE]

we have for each $a\geq\overline{a}_{\ast}$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\},$

[TABLE]

This completes the proof.

Corollary 2.2

Suppose that $N=4$ and condition $(D1)$ holds. Then for all $a\geq\overline{a}_{\ast}$, $J_{a}$ is bounded below on $H^{1}(\mathbb{R}^{N})$ and $\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)\geq 0.$

Lemma 2.3

Suppose that $N\geq 5$ and condition $(D1)$ holds. Then for all $a>0,$ $J_{a}$ is bounded below on $H^{1}(\mathbb{R}^{N})$ and there exist numbers $\widehat{r},\widehat{R}_{a}>0$ such that

[TABLE]

Furthermore, for each $0<a<\overline{a}_{\ast},$ there holds

[TABLE]

Proof. Applying the Gagliardo-Nirenberg and Young inequalities leads to

[TABLE]

where $\alpha=\frac{2^{\ast}(p-2)}{p(2^{\ast}-2)}$ and $0<\beta<\frac{p(2^{\ast}-p)}{2^{\ast}-2}.$ This implies that $J_{a}(u)$ is bounded below on $H^{1}(\mathbb{R}^{N})$ for all $a>0.$ Moreover, for each $a>0,$ there exists

[TABLE]

such that

[TABLE]

Let

[TABLE]

We now prove that

[TABLE]

Let $u\in H^{1}(\mathbb{R}^{N})$ with $\left\|u\right\|_{H^{1}}\geq\widehat{R}_{a}.$ If $\left\|u\right\|_{D^{1,2}}>R_{a},$ then the result is done clearly. If $\left\|u\right\|_{D^{1,2}}<R_{a},$ then it is enough to show that $J_{a}(u)\geq 0$ when

[TABLE]

Indeed, note that

[TABLE]

Then we have

[TABLE]

Thus, we obtain that there exists a positive number $\widehat{R}_{a}>R_{a}$ such that

[TABLE]

Moreover, using the Sobolev inequality gives

[TABLE]

which implies that there exists a number

[TABLE]

such that

[TABLE]

Hence, we have

[TABLE]

It follows from Lemma 2.1$(i)$ that for each $0<a<\overline{a}_{\ast},$

[TABLE]

Consequently, this completes the proof.

As pointed out in the section of Introduction, the Nehari manifold $\mathbf{M}_{a}$ given by

[TABLE]

can be decomposed into three parts, i.e.,

[TABLE]

Then we have the following result.

Lemma 2.4

Assume that $u_{0}$ is a local minimizer for $J_{a}$ on $\mathbf{M}_{a}$ and $u_{0}\notin\mathbf{M}_{a}^{0}.$ Then $J_{a}^{\prime}(u_{0})=0$ in $H^{-1}(\mathbb{R}^{N}).$

Proof. The proof of Lemma 2.4 is essentially same as that in Brown-Zhang [4, Theorem 2.3], so we omit it here.

Note that $u\in\mathbf{M}_{a}$ if and only if $\left\|u\right\|_{H^{1}}^{2}+a\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)^{2}-\int_{\mathbb{R}^{N}}f(x)|u|^{p}dx=0.$ By the Sobolev inequality one has

[TABLE]

which leads to

[TABLE]

Moreover, for all $u\in\mathbf{M}_{a},$ we have

[TABLE]

Thus, using $\left(\ref{2}\right)$ and $(\ref{2-2})$ gives

[TABLE]

We need the following conclusion.

Lemma 2.5

Suppose that $N\geq 1$ and $2<p<\min\{4,2^{\ast}\}.$ Then $J_{a}$ is coercive and bounded below on $\mathbf{M}_{a}^{-}.$ Furthermore, there holds

[TABLE]

Lemma 2.6

Suppose that $N=1,2,3$ and condition $(D1)$ holds. Then for each $a>0$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}$ satisfying

[TABLE]

then there exist two numbers $t_{a}^{+}$ and $t_{a}^{-}$ satisfying

[TABLE]

such that $t_{a}^{\pm}u\in\mathbf{M}_{a}^{\pm}$ and $J_{a}(t_{a}^{-}u)=\sup_{0\leq t\leq t_{a}^{+}}J_{a}(tu),$ and

[TABLE]

where $T_{f}(u)$ is defined as $(\ref{2-0}).$

Proof. Let

[TABLE]

Clearly, $tu\in\mathbf{M}_{a}$ if and only if $m(t)+a\left(\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right)^{2}=0.$ A straightforward evaluation gives

[TABLE]

Since $2<p<4$ and

[TABLE]

we find that $m(t)$ is decreasing when $0<t<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f}(u)$ and is increasing when $t>\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f}(u).$ This indicates that

[TABLE]

For each $a>0$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\}$ satisfying

[TABLE]

we can conclude that

[TABLE]

where we have used the fact of $\left(\frac{p}{2}\right)^{2/(p-2)}>1$. Moreover, by Remark 1.4 we have

[TABLE]

and a direct calculation shows that

[TABLE]

It follows from $(\ref{2-4})-(\ref{2-1})$ that

[TABLE]

Thus, there exist two numbers $t_{a}^{+},t_{a}^{-}>0$ which satisfy

[TABLE]

such that

[TABLE]

leading to $t_{a}^{\pm}u\in\mathbf{M}_{a}.$ By calculating the second order derivatives, we find

[TABLE]

and

[TABLE]

These imply that $t_{a}^{\pm}u\in\mathbf{M}_{a}^{\pm}.$ Note that

[TABLE]

Then one can see that $h_{a,u}^{\prime}(t)>0$ for all $t\in(0,t_{a}^{-})\cup(t_{a}^{+},\infty)$ and $h_{a,u}^{\prime}(t)<0$ for all $t\in(t_{a}^{-},t_{a}^{+})$. It leads to

[TABLE]

and so $J_{a}\left(t_{a}^{+}u\right)<J_{a}(t_{a}^{-}u).$ Similar to the argument of Lemma 2.1$(i),$ we have

[TABLE]

This completes the proof.

Lemma 2.7

Suppose that $N\geq 4$ and condition $(D1)$ holds. Then for each $0<a<\overline{a}_{\ast},$ there exist two numbers $t_{a}^{+}$ and $t_{a}^{-}$ satisfying

[TABLE]

such that $t_{a}^{\pm}u\in\mathbf{M}_{a}^{\pm}$ and $J_{a}(t_{a}^{-}u)=\sup_{0\leq t\leq t_{a}^{+}}J_{a}(tu)$ and $J_{a}(t_{a}^{+}u)=\inf_{t\geq t_{a}^{-}}J_{a}(tu)=\inf_{t\geq 0}J_{a}(tu)<0.$

Proof. The proof is analogous to those of Lemmas 2.1 and 2.6, so we omit it here.

Now, we follow a part of idea in [32], for any $u\in\mathbf{M}_{a}$ with $J_{a}(u)<\frac{D(p)(p-2)}{2p}\left(\frac{2S_{p}^{p}}{f_{\infty}(4-p)}\right)^{2/(p-2)},$ deduce that

[TABLE]

which implies that for $0<a<\frac{p-2}{2(4-p)}\left(\frac{4-p}{p}\right)^{2/(p-2)}\Lambda_{0},$ there exist two positive numbers $D_{1}$ and $D_{2}$ satisfying

[TABLE]

such that

[TABLE]

Thus, we obtain that

[TABLE]

where

[TABLE]

and

[TABLE]

We further have

[TABLE]

and

[TABLE]

Using the Sobolev inequality, $(\ref{2-2})$ and $(\ref{4-1})$ gives

[TABLE]

By $(\ref{4-2}),$ we derive that

[TABLE]

which implies that

[TABLE]

Using the above inequality, together with $(\ref{2-2})$, yields

[TABLE]

Hence, we have the following result.

Lemma 2.8

For $N\geq 1$ and $0<a<\frac{p-2}{2(4-p)}\left(\frac{4-p}{p}\right)^{2/(p-2)}\Lambda_{0},$ we have $\mathbf{M}_{a}^{(1)}\subset\mathbf{M}_{a}^{-}$ and $\mathbf{M}_{a}^{(2)}\subset\mathbf{M}_{a}^{+}$ both are $C^{1}$ sub-manifolds. Furthermore, each local minimizer of the functional $J_{a}$ in the sub-manifolds $\mathbf{M}_{a}^{(1)}$ and $\mathbf{M}_{a}^{(2)}$ is a critical point of $J_{a}$ in $H^{1}(\mathbb{R}^{N}).$

At the end of this section, similar to [19, Lemma 3.4], we introduce a global compactness result, which is applicable to Kirchhoff type equations.

Proposition 2.9

Suppose that $N\geq 1$ and conditions $(D1)-(D2)$ hold. Let $\{u_{n}\}$ be a bounded $(PS)_{\beta}$–sequence in $H^{1}(\mathbb{R}^{N})$ for $J_{a}.$ There exist $u_{0}\in H^{1}(\mathbb{R}^{N})$ and $A\in\mathbb{R}$ such that $I_{a}^{\prime}(u)=0,$ where

[TABLE]

*and either

$(i)$ $u_{n}\rightarrow u_{0}$ in $H^{1}(\mathbb{R}^{N}),$or

$(ii)$ there exists a number $m\in\mathbb{N}$ and $\{x_{n}^{i}\}_{n=1}^{\infty}\subset\mathbb{R}^{N}$ with $|x_{n}^{i}|\rightarrow\infty$ as $n\rightarrow\infty$ for each $1\leq i\leq m,$ nontrivial solutions $w^{1},w^{2},...,w^{m}\in H^{1}(\mathbb{R}^{N})$ of the following equation*

[TABLE]

such that

[TABLE]

and

[TABLE]

and

[TABLE]

3 Proofs of Theorems 1.1, 1.2 and

1.3

At the beginning of this section, we prove Theorem 1.1: $(i)$ For $0<k<\frac{4-N}{4-p}$ and $u\in H^{1}(\mathbb{R}^{N})\backslash\{0\},$ let $v_{t}(x)=t^{k}u(t^{-1}x).$ Then we have

$(i-A)$ $\int_{\mathbb{R}^{N}}|\nabla v_{t}(x)|^{2}dx=t^{2k-2}\int_{\mathbb{R}^{N}}|\nabla u(t^{-1}x)|^{2}dx=t^{2k-2+N}\int_{\mathbb{R}^{N}}|\nabla u(y)|^{2}dy;$

$(i-B)$ $\int_{\mathbb{R}^{N}}|v_{t}(x)|^{2}dx=t^{2k}\int_{\mathbb{R}^{N}}|u(tx)|^{2}dx=t^{2k+N}\int_{\mathbb{R}^{N}}|u(y)|^{2}dy;$

$(i-C)$ $\int_{\mathbb{R}^{N}}|v_{t}(x)|^{p}dx=t^{pk}\int_{\mathbb{R}^{N}}|u(tx)|^{p}dx=t^{pk+N}\int_{\mathbb{R}^{N}}|u(y)|^{p}dy.$

Using the above conclusions gives

[TABLE]

since $pk+N>4k-4+2N.$ This implies that $J_{a}$ is not bounded below on H^{1}(\mathbb{R}^{N})\for $N=1,2,3.$

$(ii)$ It follows from Corollary 2.2 that for each $a>\overline{a}_{\ast},$ the energy functional $J_{a}$ is bounded below on $H^{1}(\mathbb{R}^{4})$ and $\inf_{u\in H^{1}(\mathbb{R}^{4})\backslash\{0\}}J_{a}(u)>0.$ Next, we claim that for each $0<a<\underline{a}_{\ast},$ $J_{a}$ is not bounded below on $H^{1}(\mathbb{R}^{4}),$ i.e., $\inf_{u\in H^{1}(\mathbb{R}^{4})\backslash\{0\}}J_{a}(u)=-\infty.$ Let

[TABLE]

Then for $s>0,$ we have

[TABLE]

where

[TABLE]

Clearly, $I(su)=0$ if and only if $\overline{g}(s)+\frac{a}{4}\left\|u\right\|_{D^{1,2}}^{4}=0.$ It is not difficult to observe that \overline{g}\left(s_{a}\right)=0,\ \lim_{s\rightarrow 0^{+}}\overline{g}(s)=\infty\and$\ \lim_{s\rightarrow\infty}\overline{g}(s)=0,$ where

[TABLE]

Considering the derivative of $\overline{g}(s)$, we find

[TABLE]

which implies that $\overline{g}(s)$ is decreasing when $0<t<\left(\frac{p\int_{\mathbb{R}^{4}}u^{2}dx}{(4-p)f_{\min}\int_{\mathbb{R}^{4}}|u|^{p}dx}\right)^{1/(p-2)}$ and is increasing when $t>\left(\frac{p\int_{\mathbb{R}^{4}}u^{2}dx}{(4-p)f_{\min}\int_{\mathbb{R}^{4}}|u|^{p}dx}\right)^{1/(p-2)},$ and so

[TABLE]

Since $0<a<\underline{a}_{\ast},$ there exists $u\in H^{1}(\mathbb{R}^{4})\backslash\{0\}$ such that

[TABLE]

Using the above inequality, together with $(\ref{10-0}),$ leads to $\inf_{s>0}\overline{g}(s)<-\frac{a}{4}\left\|u\right\|_{D^{1,2}}^{4}.$ Set

[TABLE]

Then we obtain

[TABLE]

Let $u_{0}=s_{0}\left(u\right)u$ and $v_{t}(x)=u_{0}(t^{-1}x).$ Then we have

$(ii-A)$ $\int_{\mathbb{R}^{4}}|\nabla v_{t}(x)|^{2}dx=t^{2}\int_{\mathbb{R}^{4}}|\nabla u_{0}(x)|^{2}dx;$

$(ii-B)$ $\int_{\mathbb{R}^{4}}|v_{t}(x)|^{2}dx=t^{4}\int_{\mathbb{R}^{4}}|u_{0}(x)|^{2}dx;$

$(ii-C)$ $\int_{\mathbb{R}^{4}}|v_{t}(x)|^{p}dx=t^{4}\int_{\mathbb{R}^{4}}|u_{0}(x)|^{p}dx.$

Combining the above conclusions with $(\ref{10-11})$ gives

[TABLE]

which implies that for each $0<a<\underline{a}_{\ast},$ $J_{a}$ is not bounded below on $H^{1}(\mathbb{R}^{4}),$ i.e., $\inf_{u\in H^{1}(\mathbb{R}^{4})\backslash\{0\}}J_{a}(u)=-\infty.$

$\left(iii\right)$ By Lemmas 2.1 and 2.3, we can arrive at the conclusion.

**Next, we are ready to prove Theorem 1.2: **For $u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\},$ we know that $tu\in\mathbf{M}_{a}^{0}$ if and only if $h_{a,tu}^{\prime}\left(1\right)=h_{a,tu}^{\prime\prime}\left(1\right)=0,$ i.e., the following system of equations is satisfied:

[TABLE]

By solving the system $(\ref{15-2})$ with respect to the variables $t$ and $a,$ we have

[TABLE]

and

[TABLE]

where $\overline{A}_{f}(u)$ is as $(\ref{15-1}).$ We conclude that $a(u)$ is the unique parameter $a>0$ for which the fibering map $h_{a,u}$ has a critical point with second derivative zero at $t(u)$. Moreover, if $a>a(u)$, then $h_{a,u}$ is increasing on $(0,\infty)$ and has no critical point. Note that $\sup_{u\in H^{1}(\mathbb{R}^{N})\backslash\left\{0\right\}}a(u)=\frac{p^{2/\left(p-2\right)}}{2^{p/\left(p-2\right)}}\overline{a}_{\ast}$ by (1.5). Hence, the energy functional $J_{a}$ has no any nontrivial critical points for $a>\frac{p^{2/\left(p-2\right)}}{2^{p/\left(p-2\right)}}\overline{a}_{\ast}.$ Consequently, we complete the proof.

To prove that Theorem 1.3**, **we need the following result.

Lemma 3.1

Suppose that $N\geq 5$ and condition $(D1)$ holds. Let $0<a<\overline{a}_{\ast}$ Then every minimizing sequence for $J_{a}$ in $H^{1}(\mathbb{R}^{N})$ is bounded.

Proof. Using Lemma 2.3 gives

[TABLE]

Let $\left\{u_{n}\right\}$ be a minimizing sequence for $J_{a}$ in $H^{1}(\mathbb{R}^{N}).$ Then by Lemma 2.3 and the fact of $\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)<0$, there exists a number $\widehat{R}_{a}>0$ such that

[TABLE]

Consequently, we complete the proof.

At the end of this section, we begin to prove Theorem 1.3: $(i)$ By Lemma 3.1 and the Ekeland variational principle, for each $0<a<\overline{a}_{\ast}$ there exists a bounded minimizing sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{N})$ such that

[TABLE]

Similar to the argument of Theorem 7.1 in Appendix, we can prove that the compactness for the sequence $\{u_{n}\}$ holds. Then for each $\theta>0$ there exist a number $R=R(\theta)>0$ and a sequence $\{z_{n}\}\subset\mathbb{R}^{N}$ such that

[TABLE]

Define a new sequence of functions $v_{n}:=u_{n}(\cdot+z_{n})\in H^{1}(\mathbb{R}^{N}).$ Clearly, $\left\langle(J_{a}^{\infty})^{\prime}(v_{n}),v_{n}\right\rangle=o(1)$ and $J_{a}^{\infty}(v_{n})=\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}^{\infty}(u)+o(1).$ By virtue of $(\ref{10-2})$, for each $\theta>0$ there exists a number $R=R(\theta)>0$ such that

[TABLE]

Since $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N}),$ one can assume that there exist a subsequence $\{v_{n}\}$ and $v_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ such that

[TABLE]

By $(\ref{10-3})-(\ref{10-5})$ and Fatou’s Lemma, for any $\theta>0$ and sufficiently large $n$, there exists a number $R>0$ such that

[TABLE]

which implies that for every $p\in(2,2^{\ast}),$

[TABLE]

Since $\left\langle(J_{a}^{\infty})^{\prime}(v_{n}),v_{n}\right\rangle=o\left(1\right)$ and $\widehat{r}<\left\|v_{n}\right\|_{H^{1}}<\widehat{R}_{a},$ using $(\ref{10-6})$ gives

[TABLE]

which indicates that $v_{a}^{+}\not\equiv 0.$

Next, we show that $v_{n}\rightarrow v_{a}^{+}$ strongly in $H^{1}(\mathbb{R}^{N}).$ Suppose on the contrary. Then we have

[TABLE]

Similar to the argument of Lemma 2.6, there exists a unique $t_{a}>0$ such that

[TABLE]

where $h_{a,u}^{\infty}(t)=h_{a,u}\left(t\right)$ with $f(x)\equiv f_{\infty}$. Since $\left\langle(J_{a}^{\infty})^{\prime}(v_{n}),v_{n}\right\rangle=o(1)$, it follows from $(\ref{10-6})-(\ref{10-7})$ that

[TABLE]

Combining $(\ref{10-8})-(\ref{10-9})$ with the profile of $h_{a,v_{a}^{+}}^{\infty}(t)$ gives $t_{a}<1$. By $\left(\ref{10-6}\right)-\left(\ref{10-7}\right)$ again, we see $(h_{a,v_{n}}^{\infty})^{\prime}(t_{a})>0$ for sufficiently large $n$. Note that

[TABLE]

because of $\left\langle(J_{a}^{\infty})^{\prime}(v_{n}),v_{n}\right\rangle=o(1)$. Similar to the proof of Lemma 2.6, we obtain

[TABLE]

where

[TABLE]

One can see that $m^{\infty}(t)$ is decreasing for

[TABLE]

and

[TABLE]

This indicates that $\left(h_{a,v_{n}}^{\infty}\right)^{\prime}(t)>0$ for $0<t<1$, which implies that $h_{a,v_{n}}^{\infty}$ is increasing on $(t_{a},1)$ for sufficiently large $n$. Thus, $h_{a,v_{n}}^{\infty}(t_{a})<h_{a,v_{n}}^{\infty}(1)$ holds for sufficiently large $n$. This implies that

[TABLE]

and so, we have

[TABLE]

which is a contradiction. Thus, we obtain that $v_{n}\rightarrow v_{a}^{+}$ strongly in $H^{1}(\mathbb{R}^{N})$ and

[TABLE]

Hence, $v_{a}^{+}$ is a minimizer for $J_{a}^{\infty}$ on $H^{1}(\mathbb{R}^{N}).$ Since

[TABLE]

one can see that $v_{a}^{+}$ is a positive solution of Eq. $(E_{a})$.

Before proving Theorem 1.3 $(ii)$, we need the following compactness lemma which is regarded as a corollary of Proposition 7.1.

Lemma 3.2

Suppose that $N\geq 5$ and conditions $(D1)-(D2)$ hold. Let $\{u_{n}\}$ be a $(PS)_{\beta}$–sequence in $H^{1}(\mathbb{R}^{N})$ for $J_{a}$ with $\beta<\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}^{\infty}(u)<0.$ Then there exist a subsequence $\{u_{n}\}$ and a nonzero $u_{0}$ in $H^{1}(\mathbb{R}^{N})$ such that $u_{n}\rightarrow u_{0}$ strongly in $H^{1}(\mathbb{R}^{N})$ and $J_{a}(u_{0})=\beta.$ Furthermore, $u_{0}$ is a nonzero solution of Eq. $(E_{a}).$

The proof of Theorem 1.3 $(ii):$ By condition $(D3),$ we have

[TABLE]

Moreover, by Lemmas 2.3, 3.1$(i)$ and the Ekeland variational principle, there exists a bounded minimizing sequence $\{u_{n}\}\subset H^{1}(\mathbb{R}^{N})$ with $\widehat{r}<\left\|u_{n}\right\|_{H^{1}}<\widehat{R}_{a}$ satisfying

[TABLE]

By virtue of Lemma 3.2, we know that for each $0<a<\overline{a}_{\ast},$ Eq. $(E_{a})$ has a nontrivial solution $u_{a}^{+}$ such that $J_{a}(u_{a}^{+})=\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)<0.$ Since $J_{a}(|u_{a}^{+}|)=J_{a}(u_{a}^{+})=\inf_{u\in H^{1}(\mathbb{R}^{N})\backslash\{0\}}J_{a}(u)$, we may assume that $u_{a}^{+}$ is a positive solution of Eq. $(E_{a}).$ Consequently, we complete the proof.

4 Proofs of Theorem 1.5

In this section, we consider the following limit problem

[TABLE]

It is clear that solutions of Eq. $(E_{a}^{\infty})$ are critical points of the energy functional $J_{a}^{\infty}$ defined by

[TABLE]

Moreover, $u\in\mathbf{M}_{a}^{\infty}$ if and only if $\left\|u\right\|_{H^{1}}^{2}+a\left(\int_{\mathbb{R}^{N}}\left|\nabla u\right|^{2}dx\right)^{2}-\int_{\mathbb{R}^{N}}f_{\infty}|u|^{p}dx=0,$ where $\mathbf{M}_{a}^{\infty}=\mathbf{M}_{a}$ with $f(x)\equiv f_{\infty}.$ We denote by

[TABLE]

Since $w_{0}$ is the unique positive solution of Eq. $(E_{0}^{\infty}),$ it follows from $\left(\ref{1-8}\right)$ that

[TABLE]

for all $0<a<\frac{p-2}{2(4-p)}\left(\frac{4-p}{p}\right)^{2/(p-2)}\Lambda_{0}.$ Thus, by Lemma 2.6, there exist two numbers $t_{a}^{\infty,-},t_{a}^{\infty,+}>0$ satisfying

[TABLE]

such that $t_{a}^{\infty,\pm}w_{0}\in\mathbf{M}_{a}^{\infty,\pm},$ where $\mathbf{M}_{a}^{\infty,\pm}=\mathbf{M}_{a}^{\pm}$ with $f(x)\equiv f_{\infty}$. Furthermore, there holds

[TABLE]

and

[TABLE]

A direct calculation shows that

[TABLE]

It follows from $(\ref{28})-(\ref{27})$ that $t_{a}^{\infty,-}w_{0}\in\mathbf{M}_{a}^{\infty,(1)}$ and $t_{a}^{\infty,+}w_{0}\in\mathbf{M}_{a}^{\infty,(2)}.$ Namely, $\mathbf{M}_{a}^{\infty,\left(j\right)}(j=1,2)$ are nonempty.

Define

[TABLE]

Then we have

[TABLE]

by Lemma 2.5 and $(\ref{27}).$

We are now ready to prove Theorem 1.5: $\left(i\right)$ By Lemma 2.8 and the Ekeland variational principle, there exists a sequence $\{u_{n}\}\subset\mathbf{M}_{a}^{\infty,(1)}$ satisfies

[TABLE]

Applying Theorem 7.1 in Appendix, we obtain that for $0<a<\Lambda,$ compactness holds for the sequence $\{u_{n}\}.$ Then for each $\theta>0$ there exist a positive constant $R=R\left(\theta\right)$ and a sequence $\{z_{n}\}\subset\mathbb{R}^{N}$ such that

[TABLE]

Define a new sequence of functions $v_{n}:=u_{n}(\cdot+z_{n})\in H^{1}(\mathbb{R}^{N}).$ Then we have $\{v_{n}\}\subset\mathbf{M}_{a}^{\infty,(1)}$ and $J_{a}^{\infty}(v_{n})=\alpha_{a}^{\infty,-}+o(1).$ By virtue of $(\ref{18-4})$, for each $\theta>0$ there exists a constant $R=R(\theta)>0$ such that

[TABLE]

Since $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N}),$ one can assume that there exist a subsequence $\{v_{n}\}$ and $v_{a}^{-}\in H^{1}(\mathbb{R}^{N})$ such that

[TABLE]

In the following, by adapting the argument of Theorem 1.3 $(i),$ we obtain

[TABLE]

and

[TABLE]

Thus, $v_{a}^{-}$ is a minimizer for $J_{a}^{\infty}$ on $\mathbf{M}_{a}^{\infty,-}$ for each $0<a<\Lambda.$ By $(\ref{27})$ one has

[TABLE]

which implies that $v_{a}^{-}\in\mathbf{M}_{a}^{\infty,(1)}$. Since |v_{a}^{-}|\in\mathbf{M}_{a}^{\infty,-}\and$\ J_{a}^{\infty}(|v_{a}^{-}|)=J_{a}^{\infty}(v_{a}^{-})=\alpha_{a}^{\infty,-},$ one can see that $v_{a}^{-}$ is a positive solution of Eq. $(E_{a})$ according to Lemma 2.4. Note that for $2<p<\min\{4,2^{\ast}\},$ there holds

[TABLE]

where $\mathbf{M}_{0}^{\infty}=\mathbf{M}_{a}^{\infty}$ with $a=0$ and

[TABLE]

Then by Lemmas $\ref{g6}-\ref{g15}$, we have

[TABLE]

where $1<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f_{\infty}}(v_{a}^{-})<t_{a}^{+}$. Using the above equality, one get

[TABLE]

$(ii)$ Following the argument of Theorem 2.1 in [29]. By $(i),$ we obtain that Eq. $(E_{a})$ admits a positive solution $v_{a,1}^{-}\in H^{1}(\mathbb{R}^{N}).$ Applying Theorem 4.1 in [13] gives $v_{a,1}^{-}\rightarrow 0$ as $|x|\rightarrow\infty.$ Then after translation, we can make $v_{a,1}^{-}$ satisfy

[TABLE]

Now we show that $v_{a,1}^{-}$ is unique under $(\ref{18-5})$. Otherwise, we assume that $v_{a,2}^{-}$ is another positive solution satisfying $(\ref{18-5})$. Let

[TABLE]

Then $v_{a,i}^{-}(i=1,2)$ is a solution of the problem

[TABLE]

Let $w_{i}(x)=v_{a,i}^{-}(\sqrt{K_{i}}x).$ Then $w_{i}(x)$ is a solution of

[TABLE]

It follows from [17] that the solution of problem $(\ref{18-6})$ is unique. So $w_{1}(x)\equiv w_{2}(x)$ i.e., $v_{a,1}^{-}(\sqrt{K_{1}}x)=v_{a,2}^{-}(\sqrt{K_{2}}x).$ Thus, we have

[TABLE]

Thus, we have

[TABLE]

which implies that

[TABLE]

Define

[TABLE]

A direct calculation shows that

[TABLE]

which implies that $y(x)$ is strictly increasing when $x>b$ and $1\leq N\leq 4.$ This indicates that $K_{1}=K_{2},$since $K_{1},K_{2}>b.$ So by $(\ref{18-8})$ one have $v_{a,1}^{-}=v_{a,2}^{-}.$ Since the unique solution $w_{1}(x)$ of problem $(\ref{18-6})$ is radially symmetric by [17] and $w_{1}(x)=v_{a,1}^{-}(\sqrt{K_{1}}x)$, we obtain that $v_{a,1}^{-}$ is also radially symmetric.

$\left(iii\right)$ Note that $\Lambda\leq\overline{a}_{\ast}$ for $N\geq 5.$ Then by virtue of Theorem 1.3 $(i),$ for each $0<a<\Lambda$ Eq. $(E_{a})$ admits a positive ground state solution $v_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ such that $J_{a}^{\infty}(v_{a}^{+})<0$ when $N\geq 5.$ Clearly, $v_{a}^{+}\in\mathbf{M}_{a}^{\infty}\left(\frac{D(p)(p-2)}{2p}\left(\frac{2S_{p}^{p}}{f_{\infty}(4-p)}\right)^{2/(p-2)}\right),$ since $J_{a}^{\infty}(v_{a}^{+})<0.$ By Lemma 2.5, we further get $v_{a}^{+}\in\mathbf{M}_{a}^{\infty,(2)}.$ Thus, there holds

[TABLE]

Moreover, from $(i)$ it follows that for each $0<a<\Lambda,$ Eq. $(E_{a})$ admits a positive solution $v_{a}^{-}\in H^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Consequently, we complete the proof of Theorem 1.5.

5 Proofs of Theorems 1.6 and 1.7

By virtue of Theorem 1.5 $(i)$, we know that Eq. $(E_{a}^{\infty})$ admits a positive solution $v_{a}^{-}\in\mathbf{M}_{a}^{\infty,-}$ such that

[TABLE]

According to $(\ref{2-0}),$ one has

[TABLE]

Moreover, by Theorem 1.5 $(ii)$, we obtain that Eq. $(E_{a}^{\infty})$ admits a positive solution $v_{a}^{+}\in\mathbf{M}_{a}^{\infty,+}$ such that

[TABLE]

Similar to $(\ref{5-1})$, we have

[TABLE]

Then we have the following results.

Lemma 5.1

$(i)$* Suppose that $N\geq 1.$ Then for each $0<a<\Lambda,$ there exists $1<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f_{\infty}}(v_{a}^{-})<t_{a}^{\infty,-}$ such that*

[TABLE]

$(ii)$* Suppose that $N\geq 5.$ Then for each $0<a<\Lambda,$ there exists $0<t_{a}^{\infty,+}<1$ such that*

[TABLE]

Proof. $(i)$ Let

[TABLE]

Clearly, there holds

[TABLE]

for all $0<a<\Lambda.$ It is easy to verify that

[TABLE]

By calculating the derivative of $b_{a}^{\infty}(t)$ one has

[TABLE]

which implies that $b_{a}^{\infty}(t)$ is decreasing when $0<t<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f_{\infty}}(v_{a}^{-})$ and is increasing when $t>\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f_{\infty}}(v_{a}^{-}).$ This indicates that

[TABLE]

Moreover, we notice that

[TABLE]

Thus, it follows from $(\ref{5-4})-(\ref{5-5})$ that

[TABLE]

which shows that there exists $1<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f_{\infty}}(v_{a}^{-})<t_{a}^{\infty,-}$ such that

[TABLE]

Using a similar argument as in Lemma 2.6, we arrive at $(\ref{eqq36})$.

$(ii)$ The proof is analogous to that of part $(i)$, and we omit it here.

Lemma 5.2

Suppose that $N\geq 1$ and conditions ${(D1)}-{(D2),(D4)}-{(D5)}$ hold. Then for each $0<a<\Lambda,$ there exist two constants $t_{a}^{(1),-}$ and $t_{a}^{(2),-}$ satisfying

[TABLE]

such that $t_{a}^{(i),-}v_{a}^{-}\in\mathbf{M}_{a}^{(i)}\ (i=1,2),$ and

[TABLE]

where $T_{f}(v_{a}^{-})$ is as $(\ref{2-0})$ with $u=v_{a}^{-}.$

Proof. Let

[TABLE]

Apparently, $tv_{a}^{-}\in\mathbf{M}_{a}$ if and only if

[TABLE]

By analyzing $(\ref{eqq41}),$ we obtain

[TABLE]

A direct calculation shows that

[TABLE]

which implies that $b_{a}(t)$ is decreasing on $0<t<\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f}(v_{a}^{-})$ and is increasing on $t>\left(\frac{2}{4-p}\right)^{1/(p-2)}T_{f}(v_{a}^{-}).$ By virtue of condition $(D5)$ one has

[TABLE]

where $b_{a}^{\infty}(t)$ is given in $(\ref{5-3}).$ Using condition $(D5)$ and $(\ref{5-6})$ gives

[TABLE]

This explicitly tells us that there are two constants $t_{a}^{(1),-}$ and $t_{a}^{(2),-}$ satisfying

[TABLE]

such that

[TABLE]

That is, $t_{a}^{(i),-}v_{a}^{-}\in\mathbf{M}_{a}$ $(i=1,2).$

A direct calculation on the second order derivatives gives

[TABLE]

and

[TABLE]

So, we get $t_{a}^{(1),-}v_{a}^{-}\in\mathbf{M}_{a}^{-}$ and $t_{a}^{(2),-}v_{a}^{-}\in\mathbf{M}_{a}^{+}.$

Note that

[TABLE]

where $t_{a}^{\infty}$ is the same as described in Lemma 5.1. It follows from Lemma 5.1$(i)$ and condition $\left(D5\right)$ that for each $0<a<\Lambda,$

[TABLE]

which indicates that $t_{a}^{(1),-}v_{a}^{-}\in\mathbf{M}_{a}^{(1)}$ and $J_{a}\left(t_{a}^{(1),-}v_{a}^{-}\right)<\alpha_{a}^{\infty,-}.$ Note that

[TABLE]

Then we have $h_{a,v_{a}^{-}}^{\prime}(t)>0$ for all $t\in\left(0,t_{a}^{(1),-}\right)\cup\left(t_{a}^{(2),-},\infty\right)$ and $h_{a,v_{a}^{-}}^{\prime}(t)<0$ for all $t\in\left(t_{a}^{(1),-},t_{a}^{(2),-}\right)$. Consequently, we arrive at

[TABLE]

That is, $J_{a}\left(t_{a}^{(2),-}v_{a}^{-}\right)\leq J_{a}\left(t_{a}^{(1),-}v_{a}^{-}\right)<\alpha_{a}^{\infty,-}$, and so $t_{a}^{(2),-}v_{a}^{-}\in\mathbf{M}_{a}^{(2),-}.$ Consequently, this completes the proof.

Lemma 5.3

Suppose that $N\geq 5$ and conditions ${(D1)}-{(D4)}$ hold. Then for each $0<a<\Lambda,$ there exist two constants $t_{a}^{(1),+}$ and $t_{a}^{(2),+}$ satisfying

[TABLE]

such that $t_{a}^{(i),+}v_{a}^{+}\in\mathbf{M}_{a}^{(i)}\ (i=1,2),$ and

[TABLE]

where $T_{f}(v_{a}^{+})$ is as $(\ref{2-0})$ with $u=v_{a}^{+}.$

Proof. The proof is analogous to that of Lemma 5.2, and we omit it here.

Following [24, 35], we have the following result.

Lemma 5.4

Suppose that $0<a<\Lambda.$ Then for each $u\in\mathbf{M}_{a}^{(j)}(j=1,2),$ there exist a constant $\sigma>0$ and a differentiable function $t^{\ast}:B_{\sigma}(0)\subset H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}^{+}$ such that t^{\ast}(0)=1\and$\ t^{\ast}(v)(u-v)\in\mathbf{M}_{a}^{(j)}$ for all $v\in B_{\sigma}(0),$ and

[TABLE]

for all $\varphi\in H^{1}(\mathbb{R}^{N}).$

Proof. For any $u\in\mathbf{M}_{a}^{(j)}$, we define the function $F_{u}:\mathbb{R}\times H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ by

[TABLE]

Clearly, $F_{u}(1,0)=\left\langle J_{a}^{\prime}(u),u\right\rangle=0$ and

[TABLE]

Applying the implicit function theorem, there exist a constant $\sigma>0$ and a differentiable function $t^{\ast}:B_{\sigma}(0)\subset H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ such that $t^{\ast}(0)=1$ and

[TABLE]

for all $\varphi\in H^{1}(\mathbb{R}^{N}),$ and $F_{u}(t^{\ast}(v),v)=0$ for all $v\in B_{\sigma}(0),$ which is equivalent to

[TABLE]

According to the continuity of the map $t^{\ast}$, for $\sigma$ sufficiently small we have

[TABLE]

and

[TABLE]

Hence, $t^{\ast}(v)(u-v)\in\mathbf{M}_{a}^{(j)}$ for all $v\in B_{\sigma}(0).$ This completes the proof.

By $(\ref{4-4})$ and Lemma 2.8, we define

[TABLE]

Proposition 5.5

Suppose that $N\geq 1.$ Then for each $0<a<\Lambda,$ there exists a sequence $\{u_{n}\}\subset\mathbf{M}_{a}^{(1)}$ such that

[TABLE]

Proof. By the Ekeland variational principle [11] and Lemma 2.5, we obtain that there exists a minimizing sequence $\{u_{n}\}\subset\mathbf{M}_{a}^{(1)}$ such that

[TABLE]

and

[TABLE]

Applying Lemma 5.4 with $u=u_{n}$, there exists a function $t_{n}^{\ast}:B_{\epsilon_{n}}(0)\rightarrow\mathbb{R}$ for some $\epsilon_{n}>0$ such that $t_{n}^{\ast}(w)(u_{n}-w)\in\mathbf{M}_{a}^{(1)}.$ For $0<\delta<\epsilon_{n}$ and $u\in H^{1}(\mathbb{R}^{N})$ with $u\not\equiv 0,$ we set

[TABLE]

Since $z_{\delta}\in\mathbf{M}_{a}^{(1)},$ it follows from $(\ref{22})$ that

[TABLE]

Using the mean value theorem gives

[TABLE]

and

[TABLE]

Note that $t_{n}^{\ast}(w_{\delta})(u_{n}-w_{\delta})\in\mathbf{M}_{a}^{(1)}$. From $(\ref{23})$ it leads to

[TABLE]

We rewrite the above inequality as

[TABLE]

There exists a constant $C>0$ independent of $\delta$ such that

[TABLE]

and

[TABLE]

Letting $\delta\rightarrow 0$ in $(\ref{24})$ and using the fact that $\lim_{\delta\rightarrow 0}\left\|z_{\delta}-u_{n}\right\|_{H^{1}}=0,$ we get

[TABLE]

which leads to $(\ref{eqq20}).$

Before proving Theorem 1.6, we also need the following compactness lemma which is an immediate conclusion of Proposition 7.1.

Lemma 5.6

Suppose that $N\geq 1$ and conditions ${(D1)}-{(D2),(D4)}-{(D5)}$ hold. Let $\{u_{n}\}\subset\mathbf{M}_{a}^{(1)}$ be a $(PS)_{\beta}$–sequence in $H^{1}(\mathbb{R}^{N})$ for $J_{a}$ with $0<\beta<\alpha_{a}^{\infty,-}.$ Then there exist a subsequence $\{u_{n}\}$ and a nonzero $u_{0}$ in $H^{1}(\mathbb{R}^{N})$ such that $u_{n}\rightarrow u_{0}$ strongly in $H^{1}(\mathbb{R}^{N})$ and $J_{a}(u_{0})=\beta.$ Furthermore, $u_{0}$ is a nonzero solution of Eq. $(E_{a}).$

We are now ready to prove Theorems 1.6 and 1.7: By Proposition 5.5, there exists a sequence $\{u_{n}\}\subset\mathbf{M}_{a}^{(1)}$ satisfying

[TABLE]

It follows from Lemmas 5.6 and 5.2 that Eq. $(E_{a})$ has a nontrivial solution $u_{a}^{-}\in\mathbf{M}_{a}^{-}$ such that $J_{a}(u_{a}^{-})=\alpha_{a}^{-}.$ Thus, $u_{a}^{-}$ is a minimizer for $J_{a}$ on $\mathbf{M}_{a}^{-}.$ since

[TABLE]

one has $u_{a}^{-}\in\mathbf{M}_{a}^{(1)}$. Similarly, we obtain $|u_{a}^{-}|\in\mathbf{M}_{a}^{-}$ and $J_{a}(|u_{a}^{-}|)=J_{a}(u_{a}^{-})=\alpha_{a}^{-}$. According to Lemma 2.4, $u_{a}^{-}$ is a positive solution of Eq. $(E_{a})$ when $N\geq 1.$ Consequently, the proof of Theorem 1.6 is complete.

Note that $\Lambda\leq\overline{a}_{\ast}$ for $N\geq 5.$ Then it follows from Theorem 1.3$(ii)$ that for each $0<a<\Lambda,$ Eq. $(E_{a})$ admits a positive ground state solution $u_{a}^{+}\in H^{1}(\mathbb{R}^{N})$ such that $J_{a}(u_{a}^{+})<0.$ Clearly, $u_{a}^{+}\in\mathbf{M}_{a}\left(\frac{D(p)(p-2)}{2p}\left(\frac{2S_{p}^{p}}{f_{\infty}(4-p)}\right)^{2/(p-2)}\right),$ since $J_{a}(u_{a}^{+})<0.$ By virtue of Lemma 2.5, we further have $u_{a}^{+}\in\mathbf{M}_{a}^{(2)},$ which implies that

[TABLE]

Moreover, by Theorem 1.6 we obtain that for each $0<a<\Lambda,$ Eq. $(E_{a})$ admits a positive solution $u_{a}^{-}\in H^{1}(\mathbb{R}^{N})$ satisfying

[TABLE]

Consequently, we complete the proof of Theorem 1.7.

6 Ground State Solutions

Lemma 6.1

Suppose that $N=1$ and $f(x)\in C(\mathbb{R})$ is weakly differentiable satisfying

[TABLE]

Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then $u_{0}\in\mathbf{M}_{a}^{-}.$

Proof. Since $u_{0}$ is a nontrivial solution of Eq. $(E_{a})$, we have

[TABLE]

Following the argument of [6, Lemma 3.1], $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. $(E_{a})$ as follows

[TABLE]

Then it follows from $(\ref{3-1})-(\ref{3-4})$ and the assumption of $f(x)$ that

[TABLE]

which shows that $u_{0}\in\mathbf{M}_{a}^{-}.$ This completes the proof.

Lemma 6.2

Suppose that $N=2$ and $f(x)\in C(\mathbb{R}^{2})$ is weakly differentiable satisfying

[TABLE]

Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then $u_{0}\in\mathbf{M}_{a}^{-}.$

Proof. Since $u_{0}$ is a nontrivial solution of Eq. $(E_{a})$, there holds

[TABLE]

Moreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. $(E_{a})$ as follows

[TABLE]

Using the above two equalities gives

[TABLE]

Then it follows from $(\ref{3-3})-(\ref{3-6})$ and the assumption of $f(x)$ that

[TABLE]

which shows that $u_{0}\in\mathbf{M}_{a}^{-}.$ This completes the proof.

We are now ready to prove Theorem 1.8: Let $u^{-}$ be the positive solution of Eq. $(E_{a})$ as described in Theorem 1.5$(i)$ or 1.6$(i)$. Then there holds $u^{-}\in\mathbf{M}_{a}^{-}$ and $J_{a}(u^{-})=\inf_{u\in\mathbf{M}_{a}^{-}}J_{a}(u)=\alpha_{a}^{-}.$ By Lemma 6.1 or 6.2, $u^{-}$ is a positive ground state solution of Eq. $(E_{a}).$

Lemma 6.3

Suppose that $N=3$ and $f(x)\equiv f_{\infty}.$ Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then for each $a>0$ with $\sqrt{a^{2}+4}+\frac{2}{a}\geq A_{0},$ there holds $u_{0}\in\mathbf{M}_{a}^{-},$ where $A_{0}>0$ is defined as $(\ref{1-3}).$

Proof. Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$. Then there holds

[TABLE]

Moreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. $(E_{a})$ as follows

[TABLE]

By Azzollini [2, Theorem 1.1], for each $a>0$ there exists a constant

[TABLE]

such that $u_{0}(\cdot)=w_{0}(t_{a}\cdot).$ Then it follows from $\left(\ref{1-8}\right),(\ref{6-9})-(\ref{6-10})$ that

[TABLE]

This implies that

[TABLE]

[TABLE]

and so for each $a>0$ with $\sqrt{a^{2}+4}+\frac{2}{a}\geq A_{0},$ there holds $u_{0}\in\mathbf{M}_{a}^{-}$. This completes the proof.

Lemma 6.4

Suppose that $N=4$ and $f(x)\equiv f_{\infty}$. Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then for each $0<a\leq\overline{A}_{0},$ there holds $u_{0}\in\mathbf{M}_{a}^{-},$ where $\overline{A}_{0}>0$ is defined as $(\ref{1-4}).$

Proof. Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$. Then we have

[TABLE]

Moreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. $(E_{a})$ as follows

[TABLE]

By Azzollini [2, Theorem 1.1], for each $0<a<\left(\int_{\mathbb{R}^{4}}|\nabla w_{0}|^{2}dx\right)^{-1}$ there exists a constant

[TABLE]

such that $u_{0}(\cdot)=w_{0}(t_{a}\cdot).$ Then it follows from $(\ref{6-1})-(\ref{6-2})$ that

[TABLE]

which indicates that $u_{0}\in\mathbf{M}_{a}^{-}$ for all $0<a\leq\overline{A}_{0}.$ This completes the proof.

Lemma 6.5

Suppose that $N=4$ and $f(x)\equiv f_{\infty}$. Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then for each $a>A_{0}^{\ast},$ there holds $u_{0}\in\mathbf{M}_{a}^{+},$ where $A_{0}^{\ast}>0$ is defined as $(\ref{1-5}).$

Proof. Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ By $(\ref{6-1})-(\ref{6-2}),$ we have

[TABLE]

Applying the Gagliardo-Nirenberg and Young inequalities leads to

[TABLE]

which implies that

[TABLE]

Thus, by $(\ref{2-2})$ and $(\ref{D-1})-(\ref{D-0}),$ for each $a>A_{0}^{\ast}$ one has

[TABLE]

which indicates that $u_{0}\in\mathbf{M}_{a}^{+}.$ This complete the proof.

We are now ready to prove Theorem 1.9: By Lemmas 6.3–6.5, we can arrive at the conclusions directly.

7 Appendix

In order to verify $(\ref{18-4}),$ we use the concentration-compactness lemma [22, 23]. First of all, we discuss the three possibilities on the measures defined by a functional related to $J_{a}^{\infty}.$

For $2<p<\min\{4,2^{\ast}\},$ let $\{u_{n}\}\subset\mathbf{M}_{a}^{\infty,(1)}$ be a sequence such that

[TABLE]

Define the functional $\Phi_{a}^{\infty}:H^{1}(\mathbb{R}^{N})\rightarrow\mathbb{R}$ by

[TABLE]

By Lemma 2.5, for any $u\in\mathbf{M}_{a}^{\infty,(1)}$ one has

[TABLE]

Note that $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N}),$ since $\{u_{n}\}\subset\mathbf{M}_{a}^{\infty,(1)}.$ Then there exist a subsequence $\{u_{n}\}$ and $u_{\infty}\in H^{1}(\mathbb{R}^{N})$ such that

[TABLE]

For any $u_{n}\in\mathbf{M}_{a}^{\infty,(1)}$, we define the measure $y_{n}(\Omega)$ by

[TABLE]

Since $u_{n}\in\mathbf{M}_{a}^{\infty,\left(1\right)},$ we have $\left\|u_{n}\right\|_{H^{1}}<D_{1}.$ This implies that $y_{n}(\Omega)$ are positive measures. Furthermore, using $(\ref{7-1})$ gives

[TABLE]

and then, by P.L. Lions [22], there are three possibilities:

vanishing: for all $r>0,$

[TABLE]

where $B_{r}(\xi)=\{x\in\mathbb{R}^{N}:|x-\xi|<r\};$

dichotomy: there exist a constant $\alpha\in(0,\alpha_{a}^{\infty,-})$, two sequences $\{\xi_{n}\}$ and $\{r_{n}\},$ with $r_{n}\rightarrow+\infty$ and two nonnegative measures $y_{n}^{1}$ and $y_{n}^{2}$ such that

[TABLE]

[TABLE]

compactness: there exists a sequence $\{\xi_{n}\}\subset\mathbb{R}^{N}$ with the following property: for any $\delta>0,$ there exists $r=r(\delta)>0$ such that

[TABLE]

Theorem 7.1

For $0<a<\Lambda,$ compactness holds for the sequence of measures $\{u_{n}\}$ defined in $(\ref{7-4}).$

Proof. $(I)$ Vanishing does not occur. Suppose the contrary. Then for all $r>0,$ $(\ref{7-5})$ holds. In particular, we deduce that there exists $\bar{r}>0$ such that

[TABLE]

which implies that $u_{n}\rightarrow 0$ strongly in $L^{s}(\mathbb{R}^{N})$ for $2<s<2^{\ast}$ by using [23, Lemma I.1]. Thus, for $\{u_{n}\}\subset\mathbf{M}_{a}^{\infty,(1)},$ it follows from Lemma 2.5 that

[TABLE]

which is a contradiction.

$(II)$ Dichotomy does not occur. Suppose by contradiction that there exist a constant $\alpha\in(0,\alpha_{a}^{\infty,-})$, two sequences $\{\xi_{n}\}$ and $\{r_{n}\},$ with $r_{n}\rightarrow+\infty$ and two nonnegative measures $y_{n}^{1}$ and $y_{n}^{2}$ such that $(\ref{7-6})-(\ref{7-7})$ holds. Let $\rho_{n}\in C^{1}(\mathbb{R}^{N})$ be such that $\rho_{n}\equiv 1$ in $B_{r_{n}}(\xi_{n}),$ $\rho_{n}\equiv 0$ in $\mathbb{R}^{N}\backslash B_{2r_{n}}(\xi_{n}),$ $0\leq\rho_{n}\leq 1$ and $|\nabla\rho_{n}|\leq 2/r_{n}.$ We set

[TABLE]

It is not difficult to verify that

[TABLE]

Moreover, let us denote $\Omega_{n}:=B_{2r_{n}}(\xi_{n})\backslash B_{r_{n}}(\xi_{n}).$ Then there holds

[TABLE]

namely

[TABLE]

and

[TABLE]

A direct calculation shows that

[TABLE]

and

[TABLE]

Thus, we obtain that

[TABLE]

and

[TABLE]

Moreover, using $(\ref{7-9})$ leads to

[TABLE]

It follows from $(\ref{7-9})-\ref{7-13})$ that

[TABLE]

Thus, there holds

[TABLE]

Using the above equality, together with $(\ref{7-10})$ yields

[TABLE]

Furthermore, by $(\ref{7-9})-(\ref{7-23})$ we get

[TABLE]

We have to distinguish two cases as follows:

Case $(i):$ Up to a subsequence, $\left\langle J_{a}^{\infty})^{\prime}(h_{n}),h_{n}\right\rangle\leq 0$ or $\left\langle J_{a}^{\infty})^{\prime}(w_{n}),w_{n}\right\rangle\leq 0.$

Without loss of generality, we only consider the case of $\left\langle J_{a}^{\infty})^{\prime}(h_{n}),h_{n}\right\rangle\leq 0$, since the case of $\left\langle\left(J_{a}^{\infty}\right)^{\prime}\left(w_{n}\right),w_{n}\right\rangle\leq 0$ is similar. Note that for all $n\geq 1,$ there holds

[TABLE]

Then for each $a\leq\Lambda,$ we have

[TABLE]

By Lemmas $\ref{g6}-\ref{g15},$ for any $n\geq 1$, there exists

[TABLE]

such that $t_{a,n}^{-}h_{n}\in\mathbf{M}_{a}^{\infty,-}$, where

[TABLE]

We now prove that $t_{a,n}^{-}\leq 1.$ Suppose the contrary. Then $t_{a,n}^{-}>1.$ Since $t_{a,n}^{-}h_{n}\in\mathbf{M}_{a}^{\infty,-},$ we have

[TABLE]

Using the above equality gives

[TABLE]

Note that

[TABLE]

by $(\ref{2-2}).$ It follows from $(\ref{7-16})-(\ref{7-15})$ that

[TABLE]

which implies that

[TABLE]

However, we observe that for $2<p<\min\{4,2^{\ast}\},$

[TABLE]

This is a contradiction. Thus, $t_{a,n}^{-}\leq 1.$

Next, let us consider the functional $\Phi_{a}^{\infty}(th_{n})$ defined by

[TABLE]

A direct calculation shows that there exists a constant

[TABLE]

such that $\Phi_{a}^{\infty}(th_{n})$ is increasing on $(0,t_{a}^{\infty}(h_{n}))$ and is decreasing on $(t_{a}^{\infty}(h_{n}),\infty).$ Furthermore, by the Sobolev inequality and $(\ref{7-17})$ we have

[TABLE]

This indicates that $\Phi_{a}^{\infty}(t_{a,n}^{-}h_{n})\leq\Phi_{a}^{\infty}(h_{n}).$ Hence, for all $n\geq 1,$ there holds

[TABLE]

which is a contradiction.

Case $(ii):$ Up to a subsequence, $\left\langle(J_{a}^{\infty})^{\prime}(h_{n}),h_{n}\right\rangle>0$ and $\left\langle(J_{a}^{\infty})^{\prime}(w_{n}),w_{n}\right\rangle>0.$

By $(\ref{7-18})$ one has $\left\langle(J_{a}^{\infty})^{\prime}(h_{n}),h_{n}\right\rangle=o_{n}(1)$ and $\left\langle(J_{a}^{\infty})^{\prime}(w_{n}),w_{n}\right\rangle=o_{n}(1).$ If $t_{a,n}^{-}\leq 1+o_{n}(1),$ then we can repeat the argument of Case $(i)$ and arrive at the contradiction. Suppose that

[TABLE]

Similar to the argument of $(\ref{7-16})$, we have

[TABLE]

Similar to the argument of $(\ref{7-21}),$ one get

[TABLE]

which shows that

[TABLE]

where we have used the fact of $(t_{a,n}^{-})^{2}-\frac{2}{4-p}(t_{a,n}^{-})^{4-p}+\frac{p-2}{4-p}>0.$ Then

[TABLE]

Hence, $\Phi_{a}^{\infty}(h_{n})\rightarrow 0$ as $n\rightarrow\infty,$ which contradicts $(\ref{7-24})$. Therefore, the dichotomy cannot occur.

Acknowledgments

J. Sun is supported by the National Natural Science Foundation of China (Grant No. 11671236). T.F. Wu is supported in part by the Ministry of Science and Technology, Taiwan (Grant 108-2115-M-390-007-MY2), the Mathematics Research Promotion Center, Taiwan and the National Center for Theoretical Sciences, Taiwan.