# Multiple positive solutions for a class of Kirchhoff type equations in $$\mathbb{R}^{N}$$

## Abstract

In this paper, we study the following nonlinear Kirchhoff type equation:

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}

where a, b, V are positive constants, $$N=2$$ or 3. Under appropriate assumptions on f and h, we get that the equation has two positive solutions by using variational methods.

## 1 Introduction and main results

We consider the following nonlinear Kirchhoff type equation:

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}
(1.1)

where a, b, V are positive constants, $$N=2$$ or 3.

In recent years, the existence or multiplicity of solutions for the following Kirchhoff type equation

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+V(x)u=f(x,u),\quad x\in\mathbb{R}^{N}, \end{aligned}

where a, b are positive constants, $$N=1,2,3$$, has been widely investigated by many authors, for example [16], etc. But in those papers, the nonlinearity f satisfies 3-superlinear growth at infinity, which assures the boundedness of any Palais-Smale sequence or Cerami sequence.

Very recently, Guo [7], Li and Ye [8], Liu and Guo [9], Tang and Chen [10] studied respectively the following equation:

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+V(x)u=f(u),\quad x\in\mathbb{R}^{3}, \end{aligned}

where $$a,b$$ are positive constants, f only needs to satisfy superlinear growth at infinity. By using the Pohozaev equality, it is easy to obtain a bounded Palais-Smale sequence. Thus they obtained the existence of positive solution.

Inspired by [710], we study equation (1.1); in here, very weak conditions are assumed on f. Exactly, $$f\in C(\mathbb {R}^{+},\mathbb{R})$$ satisfies

$$(f_{1})$$ :

when $$N=2$$, there exists $$p\in(2,+\infty)$$ such that $$\lim_{t\rightarrow+\infty}\frac{f(t)}{t^{p-1}}=0$$; when $$N=3$$, $$\lim_{t\rightarrow+\infty}\frac{f(t)}{t^{5}}=0$$;

$$(f_{2})$$ :

$$\lim_{t\rightarrow0^{+}}\frac{f(t)}{t}=m\in(-\infty,V)$$;

$$(f_{3})$$ :

$$\lim_{t\rightarrow+\infty}\frac{f(t)}{t}=+\infty$$.

On h, we make the following hypotheses:

$$(h_{1})$$ :

$$h\in L^{2}(\mathbb{R}^{N})\cap C^{1}(\mathbb{R}^{N})$$ is nonnegative and $$h\not\equiv0$$;

$$(h_{2})$$ :

when $$N=2$$, $$0\leq(\nabla h(x),x)\in L^{2}(\mathbb{R}^{2})$$; when $$N=3$$, $$(\nabla h(x),x)\in L^{2}(\mathbb{R}^{3})$$;

$$(h_{3})$$ :

By using Ekeland’s variational principle [11] and Struwe’s monotonicity trick [12], we get the following.

### Theorem 1.1

Suppose that $$(f_{1})$$-$$(f_{3})$$ and $$(h_{1})$$-$$(h_{3})$$ hold. Then there exists $$m_{0}>0$$ such that, when $$(\int_{\mathbb{R}^{N}}h^{2}\,dx )^{\frac{1}{2}}< m_{0}$$, equation (1.1) has two positive solutions.

When $$f(t)<0$$, by $$(f_{2})$$ and $$(f_{3})$$, there exists $$l>0$$ such that $$f(t)+lt\geq0$$ for all $$t\geq0$$. Thus equation (1.1) is equivalent to the following equation:

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Wu=k(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}
(1.2)

where $$W=V+l>0$$ and $$k(t)=f(t)+lt\in C(\mathbb{R}^{+},\mathbb{R}^{+})$$ satisfies

$$(k_{1})$$ :

when $$N=2$$, there exists $$p\in(2,+\infty)$$ such that $$\lim_{t\rightarrow+\infty}\frac{k(t)}{t^{p-1}}=0$$; when $$N=3$$, $$\lim_{t\rightarrow+\infty}\frac{k(t)}{t^{5}}=0$$;

$$(k_{2})$$ :

$$\lim_{t\rightarrow0^{+}}\frac{k(t)}{t}=m+l:=d\in[0,W)$$;

$$(k_{3})$$ :

$$\lim_{t\rightarrow+\infty}\frac{k(t)}{t}=+\infty$$.

Hence in order to prove Theorem 1.1, we only need to prove the following.

### Theorem 1.2

Suppose that $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$-$$(h_{3})$$ hold. Then there exists $$m_{0}>0$$ such that when $$(\int_{\mathbb{R}^{N}}h^{2}\,dx )^{\frac {1}{2}}< m_{0}$$, equation (1.2) has two positive solutions.

### Remark 1.3

Under hypotheses on k, we are not able to obtain directly the boundedness of the Palais-Smale sequences. Inspired by Jeanjean’s idea in [13] and [14], we will use an indirect approach, i.e., Struwe’s monotonicity trick developed by Jeanjean. It is worth pointing out that comparing with $$N=3$$, when $$N=2$$, it is more complex to prove the boundedness of the Palais-Smale sequences, which will be seen in Lemma 3.8.

## 2 Preliminaries

From now on, we will use the following notations.

• $$E:=\{u\in H^{1}(\mathbb{R}^{N}):u(x)=u( \vert x \vert )\}$$ is the usual Sobolev space endowed with the norm

\begin{aligned} \Vert u \Vert = \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}+u^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}
• $$D^{1,2}(\mathbb{R}^{N})$$ is completion of $$C_{0}^{\infty}(\mathbb {R}^{N})$$ with respect to the norm

\begin{aligned} \Vert u \Vert _{D^{1,2}(\mathbb{R}^{N})}= \biggl( \int_{\mathbb {R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}
• For any $$1\leq p<\infty$$, $$L^{p}(\mathbb{R}^{N})$$ denotes the Lebesgue space and its norm is denoted by

\begin{aligned} \vert u \vert _{p}= \biggl( \int_{\mathbb{R}^{N}} \vert u \vert ^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}
• $$\langle\cdot,\cdot\rangle$$ denotes the action of dual, $$(\cdot,\cdot)$$ denotes the inner product in $$\mathbb{R}^{N}$$.

• C, $$C_{i}$$ denote various positive constants.

Since we are looking for positive solution, we may assume that $$k(t)=0$$ for all $$t<0$$. Under the assumptions on k and h, it is obvious that the functional $$I:E\rightarrow\mathbb{R}$$ defined by

\begin{aligned} I(u)=\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx+ \frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}u^{2}\,dx- \int_{\mathbb {R}^{N}}K(u)\,dx- \int_{\mathbb{R}^{N}}hu\,dx \end{aligned}

is of class $$C^{1}$$, where $$K(t)=\int_{0}^{t}k(s)\,ds$$ and

\begin{aligned} \bigl\langle I'(u),v\bigr\rangle =& \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \int_{\mathbb{R}^{N}}(\nabla u,\nabla v)\,dx+W \int _{\mathbb{R}^{N}}uv\,dx- \int_{\mathbb{R}^{N}}k(u)v\,dx \\ &{}- \int_{\mathbb{R}^{N}}hv\,dx, \end{aligned}

for all $$u, v\in E$$. As is well known, the weak solution of equation (1.2) is the critical point of I in E.

## 3 Proof of the main results

Next lemma can be viewed as a generalization of Struwe’s monotonicity trick [12] and is the main tool for obtaining a bounded Palais-Smale sequence.

### Lemma 3.1

(see [13] or [14])

Let X be a Banach space equipped with a norm $$\Vert \cdot \Vert _{X}$$, and let $$J\subset\mathbb{R}^{+}$$ be an interval. We consider a family $$\{\Phi_{\mu}\}_{\mu\in J}$$ of $$C^{1}$$-functionals on X of the form

$$\Phi_{\mu}(u)=A(u)-\mu B(u),\quad\forall\mu\in J,$$

where $$B(u)\geq0$$ for all $$u\in X$$ and such that either $$A(u)\rightarrow +\infty$$ or $$B(u)\rightarrow+\infty$$ as $$\Vert u \Vert _{X}\rightarrow+\infty$$. We assume that there are two points $$v_{1}$$, $$v_{2}$$ in X such that

$$c_{\mu}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]} \Phi_{\mu}\bigl(\gamma(t)\bigr)>\max\bigl\{ \Phi_{\mu}(v_{1}), \Phi_{\mu}(v_{2})\bigr\} ,$$

where

$$\Gamma=\bigl\{ \gamma\in C\bigl([0,1],X\bigr):\gamma(0)=v_{1}, \gamma(1)=v_{2}\bigr\} .$$

Then, for almost every $$\mu\in J$$, there is a bounded $$(PS)_{c_{\mu}}$$ sequence for $$\Phi_{\mu}$$, that is, there exists a sequence $$\{u_{n}\}\subset X$$ such that

1. (1)

$$\{u_{n}\}$$ is bounded in X,

2. (2)

$$\Phi_{\mu}(u_{n})\rightarrow c_{\mu}$$,

3. (3)

$$\Phi'_{\mu}(u_{n})\rightarrow0$$ in $$X^{*}$$, where $$X^{*}$$ is the dual of X.

### Remark 3.2

In [13], it is also proved that, under the assumptions of Lemma 3.1, the map $$\mu\mapsto c_{\mu}$$ is left-continuous.

In the paper, we set $$X:=E$$, $$\Vert \cdot \Vert _{X}:= \Vert \cdot \Vert$$ and $$J:=[\frac{1}{2},1]$$. Let us define $$I_{\mu}:E\rightarrow\mathbb{R}$$ by $$I_{\mu}(u)=A(u)-\mu B(u)$$, where

\begin{aligned}& A(u)=\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx+ \frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}u^{2}\,dx- \int_{\mathbb{R}^{N}}hu\,dx, \\& B(u)= \int_{\mathbb{R}^{N}}K(u)\,dx. \end{aligned}

Then $$I_{1}(u)=I(u)$$. By $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$, it is obvious that $$I_{\mu}\in C^{1}(E,\mathbb{R})$$, $$B(u)\geq0$$ for all $$u\in E$$ and $$A(u)\geq\frac{\min\{a,W\}}{2} \Vert u \Vert ^{2}-C \vert h \vert _{2} \Vert u \Vert \rightarrow+\infty$$ as $$\Vert u \Vert \rightarrow+\infty$$.

### Lemma 3.3

Assume that $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$ hold. Then there exist $$\rho >0$$, $$\alpha>0$$ and $$m_{0}>0$$ such that $$I_{\mu}(u)\vert_{\Vert u\Vert=\rho}\geq\alpha$$ for all h satisfying $$\vert h\vert_{2}< m_{0}$$ and for all $$\mu\in J$$.

### Proof

First, we consider $$N=2$$. It follows from $$(k_{1})$$ and $$(k_{2})$$ that, for all $$t\in\mathbb{R}$$, we have

\begin{aligned} \bigl\vert K(t) \bigr\vert \leq\frac{W+d}{4} \vert t \vert ^{2}+C \vert t \vert ^{p}. \end{aligned}
(3.1)

By (3.1), the Hölder inequality and the Sobolev inequality, for all $$\mu\in J$$ and $$u\in E$$, one has

\begin{aligned} I_{\mu}(u) \geq&\frac{a}{2} \int_{\mathbb{R}^{2}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{2}}u^{2}\,dx- \int_{\mathbb {R}^{2}}K(u)\,dx- \int_{\mathbb{R}^{2}}hu\,dx \\ \geq&\frac{a}{2} \int_{\mathbb{R}^{2}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{2}}u^{2}\,dx-\frac{W+d}{4} \int_{\mathbb {R}^{2}}u^{2}\,dx-C \int_{\mathbb{R}^{2}} \vert u \vert ^{p}\,dx- \vert h \vert _{2} \vert u \vert _{2} \\ \geq&\frac{\min\{2a,W-d\}}{4} \Vert u \Vert ^{2}-C_{1} \Vert u \Vert ^{p}-C_{2} \vert h \vert _{2} \Vert u \Vert \\ =& \Vert u \Vert \biggl(\frac{\min\{2a,W-d\}}{4} \Vert u \Vert -C_{1} \Vert u \Vert ^{p-1}-C_{2} \vert h \vert _{2} \biggr). \end{aligned}

Let $$g_{1}(t)=\frac{\min\{2a,W-d\}}{4}t-C_{1}t^{p-1}$$ for $$t\geq0$$. Since $$p>2$$, we know that there exists a constant $$\rho>0$$ such that $$\max_{t\geq 0}g_{1}(t)=g_{1}(\rho)>0$$. Choose $$m_{0}=\frac{1}{2C_{2}}g_{1}(\rho)$$, then there exists $$\alpha>0$$ such that $$I_{\mu}(u)\vert_{\Vert u\Vert=\rho}\geq\alpha$$ for all h satisfying $$\vert h\vert_{2}< m_{0}$$.

Next when $$N=3$$, it follows from $$(k_{1})$$ and $$(k_{2})$$ that, for all $$t\in\mathbb{R}$$, we have

\begin{aligned} \bigl\vert K(t) \bigr\vert \leq\frac{W+d}{4} \vert t \vert ^{2}+C \vert t \vert ^{6}. \end{aligned}
(3.2)

By (3.2), the Hölder inequality and the Sobolev inequality, for all $$\mu\in J$$ and $$u\in E$$, one has

\begin{aligned} I_{\mu}(u) \geq&\frac{a}{2} \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{3}}u^{2}\,dx- \int_{\mathbb {R}^{3}}K(u)\,dx- \int_{\mathbb{R}^{3}}hu\,dx \\ \geq&\frac{a}{2} \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{3}}u^{2}\,dx-\frac{W+d}{4} \int_{\mathbb {R}^{3}}u^{2}\,dx-C \int_{\mathbb{R}^{3}} \vert u \vert ^{6}\,dx- \vert h \vert _{2} \vert u \vert _{2} \\ \geq&\frac{\min\{2a,W-d\}}{4} \Vert u \Vert ^{2}-C_{3} \Vert u \Vert ^{6}-C_{4} \vert h \vert _{2} \Vert u \Vert \\ =& \Vert u \Vert \biggl(\frac{\min\{2a,W-d\}}{4} \Vert u \Vert -C_{3} \Vert u \Vert ^{5}-C_{4} \vert h \vert _{2} \biggr). \end{aligned}

Let $$g_{2}(t)=\frac{\min\{2a,W-d\}}{4}t-C_{3}t^{5}$$ for $$t\geq0$$, we know that there exists a constant $$\rho>0$$ such that $$\max_{t\geq0}g_{2}(t)=g_{2}(\rho)>0$$. Choose $$m_{0}=\frac {1}{2C_{4}}g_{2}(\rho)$$, then there exists $$\alpha>0$$ such that $$I_{\mu}(u)\vert_{\Vert u\Vert =\rho}\geq\alpha$$ for all h satisfying $$\vert h\vert_{2}< m_{0}$$. □

### Lemma 3.4

Assume that $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$ hold. Then $$-\infty< c:=\inf\{ I(u): \Vert u \Vert \leq\rho\}<0$$, where ρ is given by Lemma 3.3.

### Proof

Since $$h\in L^{2}(\mathbb{R}^{N})$$ and $$h\not\equiv0$$, then for $$\varepsilon=\frac{ \vert h \vert _{2}}{2}$$, there exists $$\phi \in C^{\infty}_{0}(\mathbb{R}^{N})$$ such that $$\vert h-\phi \vert _{2}<\varepsilon$$. Thus

\begin{aligned} \int_{\mathbb{R}^{N}}\bigl(h^{2}-h\phi\bigr)\,dx\leq \int_{\mathbb {R}^{N}} \bigl\vert h^{2}-h\phi \bigr\vert \,dx \leq \vert h-\phi \vert _{2} \vert h \vert _{2}< \varepsilon \vert h \vert _{2}, \end{aligned}

and then

\begin{aligned} \int_{\mathbb{R}^{N}}h\phi \,dx\geq \vert h \vert _{2}^{2}- \varepsilon \vert h \vert _{2}=\frac{ \vert h \vert _{2}^{2}}{2}>0. \end{aligned}

Hence

\begin{aligned} I(t\phi)\leq\frac{at^{2}}{2} \int_{\mathbb{R}^{N}} \vert \nabla\phi \vert ^{2}\,dx+ \frac{bt^{4}}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla\phi \vert ^{2}\,dx \biggr)^{2}+\frac{Wt^{2}}{2} \int_{\mathbb {R}^{N}}\phi^{2}\,dx-t \int_{\mathbb{R}^{N}}h\phi \,dx< 0 \end{aligned}

for $$t>0$$ small enough. Then we get $$c=\inf\{I(u): \Vert u \Vert \leq\rho\}<0$$. $$c>-\infty$$ is obvious. □

In order to prove the compactness, we define $$g(t)=k(t)-dt$$, $$\forall t\in\mathbb{R}$$. Then, by $$(k_{1})$$ and $$(k_{2})$$, we get that

\begin{aligned} \lim_{t\rightarrow0^{+}}\frac{g(t)}{t}=0, \end{aligned}
(3.3)

and when $$N=2$$,

\begin{aligned} \lim_{t\rightarrow+\infty}\frac{g(t)}{t^{p-1}}=0, \end{aligned}
(3.4)

when $$N=3$$,

\begin{aligned} \lim_{t\rightarrow+\infty}\frac{g(t)}{t^{5}}=0. \end{aligned}
(3.5)

### Lemma 3.5

Suppose that $$(k_{1})$$-$$(k_{3})$$, $$(h_{1})$$ and $$(h_{3})$$ hold. Assume that $$\{ u_{n}\}\subset E$$ is a bounded Palais-Smale sequence of $$I_{\mu}$$ for each $$\mu\in J$$. Then $$\{u_{n}\}$$ has a convergent subsequence in E.

### Proof

Since $$\{u_{n}\}$$ is bounded in E and $$E\hookrightarrow L^{s}(\mathbb{R}^{3})$$, $$\forall s\in(2,6)$$, $$E\hookrightarrow L^{s}(\mathbb {R}^{2})$$, $$\forall s\in(2,+\infty)$$ are compact (see [15]), up to a subsequence, we can assume that there exists $$u\in E$$ such that $$u_{n}\rightharpoonup u$$ in E, $$u_{n}\rightarrow u$$ in $$L^{s}(\mathbb{R}^{3})$$, $$\forall s\in(2,6)$$, $$u_{n}\rightarrow u$$ in $$L^{s}(\mathbb{R}^{2})$$, $$\forall s\in(2,+\infty)$$, $$u_{n}(x)\rightarrow u(x)$$ a.e. in $$\mathbb{R}^{N}$$.

By (3.3) and (3.4), for any $$\varepsilon>0$$, we have

\begin{aligned} \bigl\vert g(t) \bigr\vert \leq\varepsilon \vert t \vert +C_{\varepsilon} \vert t \vert ^{p-1},\quad \forall t\geq0. \end{aligned}
(3.6)

Then, by (3.6) and the Hölder inequality, one has

\begin{aligned}& \biggl\vert \int_{\mathbb{R}^{2}}g(u_{n}) (u_{n}-u)\,dx \biggr\vert \\& \quad\leq\varepsilon \int_{\mathbb{R}^{2}} \vert u_{n} \vert \vert u_{n}-u \vert \,dx+C_{\varepsilon}\int_{\mathbb{R}^{2}} \vert u_{n} \vert ^{p-1} \vert u_{n}-u \vert \,dx \\& \quad\leq\varepsilon \vert u_{n} \vert _{2} \vert u_{n}-u \vert _{2}+C_{\varepsilon}\biggl( \int_{\mathbb{R}^{2}} \vert u_{n} \vert ^{p}\,dx \biggr)^{\frac{p-1}{p}} \vert u_{n}-u \vert _{p} \\& \quad\leq C\varepsilon+o_{n}(1). \end{aligned}

Similarly, we can obtain that

\begin{aligned} \biggl\vert \int_{\mathbb{R}^{2}}g(u) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}

By (3.3) and (3.5), for any $$\varepsilon>0$$, we have

\begin{aligned} \bigl\vert g(t) \bigr\vert \leq\varepsilon\bigl( \vert t \vert + \vert t \vert ^{5}\bigr)+C_{\varepsilon} \vert t \vert ^{3},\quad \forall t\geq0. \end{aligned}
(3.7)

Hence, by (3.7) and the Hölder inequality, one has

\begin{aligned}& \biggl\vert \int_{\mathbb{R}^{3}}g(u_{n}) (u_{n}-u)\,dx \biggr\vert \\& \quad\leq\varepsilon \int_{\mathbb{R}^{3}} \vert u_{n} \vert \vert u_{n}-u \vert \,dx+\varepsilon \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{5} \vert u_{n}-u \vert \,dx+C_{\varepsilon}\int _{\mathbb{R}^{3}} \vert u_{n} \vert ^{3} \vert u_{n}-u \vert \,dx \\& \quad\leq\varepsilon \vert u_{n} \vert _{2} \vert u_{n}-u \vert _{2}+\varepsilon \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{6}\,dx \biggr)^{\frac{5}{6}} \vert u_{n}-u \vert _{6} +C_{\varepsilon}\biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{\frac {9}{2}}\,dx \biggr)^{\frac{2}{3}} \vert u_{n}-u \vert _{3} \\& \quad\leq C\varepsilon+o_{n}(1). \end{aligned}

Similarly, we can obtain that

\begin{aligned} \biggl\vert \int_{\mathbb{R}^{3}}g(u) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}

Hence when $$N=2$$ or 3, one has

\begin{aligned} \biggl\vert \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}

It is clear that

$$\bigl\langle I'_{\mu}(u_{n})-I'_{\mu}(u),u_{n}-u \bigr\rangle =o_{n}(1)$$

and

$$b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u) \bigr)\,dx=o_{n}(1).$$

Note that

\begin{aligned} \bigl\langle I_{\mu}'(u_{n})-I'_{\mu}(u),u_{n}-u \bigr\rangle =& \biggl(a+b \int_{\mathbb {R}^{N}} \vert \nabla u_{n} \vert ^{2}\,dx \biggr) \int_{\mathbb {R}^{N}} \bigl\vert \nabla(u_{n}-u) \bigr\vert ^{2}\,dx \\ &{}+(W-\mu d) \int_{\mathbb{R}^{N}} \vert u_{n}-u \vert ^{2}\,dx \\ &{}-b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u)\bigr)\,dx \\ &{}-\mu \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx \\ \geq& \min\{a,W-\mu d\} \Vert u_{n}-u \Vert ^{2} \\ &{}-b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u)\bigr)\,dx \\ &{}-\mu \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx. \end{aligned}

Therefore we get that $$\Vert u_{n}-u \Vert \rightarrow0$$ as $$n\rightarrow\infty$$. □

### Proof of the first solution of Theorem 1.2

By Lemma 3.4 and Ekeland’s variational principle [11], there exists a sequence $$\{u_{n}\}\subset E$$ such that $$\Vert u_{n} \Vert \leq\rho$$, $$I(u_{n})\rightarrow c$$ and $$I'(u_{n})\rightarrow 0$$ as $$n\rightarrow\infty$$. From Lemma 3.5 with $$\mu=1$$, there exists $$u_{0}\in E$$ such that $$u_{n}\rightarrow u_{0}$$ in E and then $$I'(u_{0})=0$$ and $$I(u_{0})=c<0$$. Put $$u_{0}^{-}:=\max\{-u_{0},0\}$$, one has

\begin{aligned}[b] 0&=\bigl\langle I'(u_{0}),u_{0}^{-} \bigr\rangle \\ &=-a \int_{\mathbb{R}^{N}} \bigl\vert \nabla u_{0}^{-} \bigr\vert ^{2}\,dx-b \int _{\mathbb{R}^{N}} \vert \nabla u_{0} \vert ^{2}\,dx \int_{\mathbb {R}^{N}} \bigl\vert \nabla u_{0}^{-} \bigr\vert ^{2}\,dx-W \int_{\mathbb{R}^{N}} \bigl\vert u_{0}^{-} \bigr\vert ^{2}\,dx \\ &\quad{}- \int_{\mathbb{R}^{N}}hu_{0}^{-}dx, \end{aligned}
(3.8)

which implies that $$u_{0}^{-}=0$$ and then $$u_{0}\geq0$$. By the strong maximum principle, we get $$u_{0}>0$$. □

For ρ and α in Lemma 3.3, we have following result.

### Lemma 3.6

Assume that $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$ hold. Then

$$(*)$$ :

$$\exists v_{2}\in E$$ with $$\Vert v_{2} \Vert >\rho$$ such that $$I_{\mu}(v_{2})<0$$, $$\forall\mu\in J$$.

$$(**)$$ :

$$c_{\mu}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I_{\mu}(\gamma (t))>\max\{I_{\mu}(0),I_{\mu}(v_{2})\}$$, $$\forall\mu\in J$$, where

\begin{aligned} \Gamma=\bigl\{ \gamma\in C\bigl([0,1],E\bigr):\gamma(0)=0,\gamma(1)=v_{2} \bigr\} . \end{aligned}

### Proof

It follows from $$(k_{3})$$ that, for any $$L>0$$, there exists $$C_{L}>0$$ such that, for all $$t\geq0$$, one has

$$K(t)\geq Lt^{2}-C_{L}.$$
(3.9)

Fix $$0\leq w\in C_{0}^{\infty}(\mathbb{R}^{N})$$ with $$\operatorname {supp}w\subset B_{1}:=\{x\in\mathbb{R}^{N}: \vert x \vert <1\}$$ and $$w\not\equiv0$$. Define $$w_{t}(x)=tw(\frac{x}{t^{2}})$$ for $$t>0$$, then

\begin{aligned} \operatorname{supp}w_{t}=\bigl\{ t^{2}y: y\in\operatorname{supp}w\bigr\} . \end{aligned}

By direct computation, we have

\begin{aligned}& \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx=t^{2N-2} \int _{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx, \\& \int_{\mathbb{R}^{N}}w_{t}^{2}\,dx=t^{2N+2} \int_{\mathbb{R}^{N}}w^{2}\,dx \end{aligned}

and, by (3.9),

\begin{aligned} \int_{\mathbb{R}^{N}}K(w_{t})\,dx =& \int_{\operatorname{supp}w_{t}}K(w_{t})\,dx \\ \geq& L \int_{\operatorname{supp}w_{t}}w_{t}^{2}\,dx-C_{L} \int_{\operatorname{supp}w_{t}}\,dx \\ \geq& Lt^{2N+2} \int_{\operatorname{supp}w}w^{2}\,dx-C_{L} \int_{\{t^{2}y:y\in B_{1}\} }\,dx \\ =&Lt^{2N+2} \int_{\mathbb{R}^{N}}w^{2}\,dx-C_{L}Ct^{2N}. \end{aligned}

Therefore

\begin{aligned}& I_{\mu}(w_{t}) \\& \quad =\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx+\frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}w_{t}^{2}\,dx\\& \qquad {}-\mu \int _{R^{N}}K(w_{t})\,dx- \int_{\mathbb{R}^{N}}hw_{t}\,dx \\& \quad \leq\frac{at^{2N-2}}{2} \int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx+ \frac{bt^{4N-4}}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx \biggr)^{2}+\frac{Wt^{2N+2}}{2} \int_{\mathbb {R}^{N}}w^{2}\,dx \\& \qquad {}-\frac{Lt^{2N+2}}{2} \int_{\mathbb{R}^{N}}w^{2}\,dx+ C_{L}Ct^{2N} \end{aligned}

for all $$\mu\in J$$. When $$N=2$$, we choose $$L=2W$$. When $$N=3$$, we choose $$L=2W+b\frac{ (\int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx )^{2}}{\int_{\mathbb{R}^{N}}w^{2}\,dx}$$. Then $$I_{\mu}(w_{t})\rightarrow -\infty$$ as $$t\rightarrow+\infty$$. Hence there exists $$t'>0$$ such that $$v_{2}:=w_{t'}$$ with $$\Vert v_{2} \Vert >\rho$$ and $$I_{\mu}(v_{2})<0$$, $$\forall\mu\in J$$. This completes the proof of $$(*)$$.

By Lemma 3.3 and the definition of $$c_{\mu}$$, for all $$\mu\in J$$, we have

$$0< \alpha\leq c_{1}\leq c_{\mu}\leq c_{\frac{1}{2}}\leq\max _{t\in [0,1]}I_{\frac{1}{2}}(tv_{2})< +\infty.$$

Therefore, by $$I_{\mu}(0)=0$$ and $$I_{\mu}(v_{2})<0$$, we obtain the proof of $$(**)$$. □

So far we have verified all the conditions of Lemma 3.1. Then there exists $$\{\mu_{j}\}\subset J$$ such that

1. (i)

$$\mu_{j}\rightarrow1^{-}$$ as $$j\rightarrow\infty$$, $$\{u_{n}^{j}\}$$ is bounded in E;

2. (ii)

$$I_{\mu_{j}}(u_{n}^{j})\rightarrow c_{\mu_{j}}$$ as $$n\rightarrow \infty$$;

3. (iii)

$$I'_{\mu_{j}}(u_{n}^{j})\rightarrow0$$ as $$n\rightarrow\infty$$.

Using (i)-(iii) and Lemma 3.5, there exists $$u_{j}\in E$$ such that $$u_{n}^{j}\rightarrow u_{j}$$ in E as $$n\rightarrow\infty$$ and then $$I_{\mu_{j}}(u_{j})=c_{\mu_{j}}$$ and $$I'_{\mu_{j}}(u_{j})=0$$. Hence, from $$I_{\mu_{j}}(u_{j})=c_{\mu_{j}}$$ and $$\langle I'_{\mu _{j}}(u_{j}),u_{j}\rangle=0$$, we get respectively

\begin{aligned}& \begin{aligned}[b] &\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}u_{j}^{2}\,dx \\ &\quad-\mu_{j} \int _{\mathbb{R}^{N}}K(u_{j})\,dx- \int_{\mathbb{R}^{N}}hu_{j}\,dx=c_{\mu_{j}}, \end{aligned} \end{aligned}
(3.10)
\begin{aligned}& \begin{aligned}[b] & a \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx+b \biggl( \int _{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+W \int _{\mathbb{R}^{N}}u_{j}^{2}\,dx \\ &\quad-\mu_{j} \int_{\mathbb{R}^{N}}k(u_{j})u_{j}\,dx- \int_{\mathbb {R}^{N}}hu_{j}\,dx=0. \end{aligned} \end{aligned}
(3.11)

Next, for obtaining $$\{u_{j}\}$$ is bounded in E, we need the following lemma (Pohozaev type identity). The proof is similar to Lemma 2.6 in [16], and we omit its proof in here.

### Lemma 3.7

Suppose that $$(h_{1})$$ and $$(h_{2})$$ hold. If $$I'_{\mu}(u)=0$$, we have

\begin{aligned} &\frac{a(N-2)}{2} \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx+ \frac{b(N-2)}{2} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{NW}{2} \int_{\mathbb{R}^{N}}u^{2}\,dx \\ &\quad{}-N\mu \int_{\mathbb{R}^{N}}K(u)\,dx-N \int_{\mathbb{R}^{N}}hu\,dx- \int _{\mathbb{R}^{N}}\bigl(\nabla h(x),x\bigr)u\,dx=0. \end{aligned}

Since $$I'_{\mu_{j}}(u_{j})=0$$, by Lemma 3.7, we get that

\begin{aligned}[b] &\frac{a(N-2)}{2} \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac{b(N-2)}{2} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+\frac{NW}{2} \int_{\mathbb{R}^{N}}u_{j}^{2}\,dx \\ &\quad{}-N\mu_{j} \int_{\mathbb{R}^{N}}K(u_{j})\,dx-N \int_{\mathbb {R}^{N}}hu_{j}\,dx- \int_{\mathbb{R}^{N}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx=0. \end{aligned}
(3.12)

### Lemma 3.8

Assume that $$(k_{1})$$-$$(k_{3})$$ and $$(h_{1})$$-$$(h_{3})$$ hold. Then $$\{u_{j}\}$$ is bounded in E.

### Proof

It follows from (3.10) and (3.12) that

$$a \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b(4-N)}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+ \int_{\mathbb{R}^{N}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx=Nc_{\mu_{j}}.$$
(3.13)

Be similar to (3.8), by $$I'_{\mu_{j}}(u_{j})=0$$, we obtain $$u_{j}\geq0$$.

Firstly, we consider $$N=2$$. From (3.13) and $$c_{\mu_{j}}\leq c_{\frac{1}{2}}$$, we get

\begin{aligned}[b] a \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx &\leq a \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{2} \biggl( \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2} \\ &\quad{}-2c_{\mu_{j}}+2c_{\mu_{j}} \\ &=- \int_{\mathbb{R}^{2}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx+2c_{\mu_{j}}. \end{aligned}
(3.14)

Since $$(\nabla h(x),x)\geq0$$, by (3.14) and $$u_{j}\geq0$$, one has $$\{\int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx\}$$ is bounded. Next we prove $$\{\int_{\mathbb{R}^{2}}u_{j}^{2}\,dx\}$$ is bounded. Inspired by [14], we suppose by contradiction that $$\lambda _{j}:= \vert u_{j} \vert _{2}\rightarrow+\infty$$. Define $$w_{j}:=u_{j}(\lambda_{j}x)$$, then

$$\int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx= \int_{\mathbb {R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx\leq C$$

and

$$\int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx= \frac{1}{\lambda _{j}^{2}} \int_{\mathbb{R}^{2}} \vert u_{j} \vert ^{2}\,dx=1.$$
(3.15)

Hence $$\{w_{j}\}$$ is bounded in E. Up to a subsequence, we may assume that $$w_{j}\rightharpoonup w$$ in E, $$w_{j}\rightarrow w$$ in $$L^{s}(\mathbb {R}^{2})$$, $$\forall s\in(2,+\infty)$$, $$w_{j}\rightarrow w$$ in $$L_{\mathrm{loc}}^{s}(\mathbb{R}^{2})$$, $$\forall s\in[1,+\infty)$$, $$w_{j}(x)\rightarrow w(x)$$ a.e. in $$\mathbb{R}^{2}$$. By $$I'_{\mu_{j}}(u_{j})=0$$, one has

$$- \biggl(a+b \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx \biggr)\frac{1}{\lambda_{j}^{2}}\triangle w_{j}+(W-d \mu_{j})w_{j}=\mu_{j}g(w_{j})+h( \lambda_{j}x).$$
(3.16)

For any $$v\in C_{0}^{\infty}(\mathbb{R}^{2})$$, one has

$$\biggl\vert \int_{\mathbb{R}^{2}}h(\lambda_{j}x)v\,dx \biggr\vert \leq \vert v \vert _{2} \biggl( \int_{\mathbb{R}^{2}} \bigl\vert h(\lambda _{j}x) \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} =\frac{1}{\lambda_{j}} \vert v \vert _{2} \vert h \vert _{2}\rightarrow0$$
(3.17)

and by the Lebesgue dominated convergence theorem, we have

$$\biggl\vert \int_{\mathbb{R}^{2}}g(w_{j})v\,dx- \int_{\mathbb{R}^{2}}g(w)v\,dx \biggr\vert \leq C \int_{\operatorname{supp}v} \bigl\vert g(w_{j})-g(w) \bigr\vert \,dx \rightarrow0.$$
(3.18)

Hence by (3.16)-(3.18), we have $$(W-d)w=g(w)$$ in $$\mathbb{R}^{2}$$, from which we get that $$w=0$$. Indeed, since 0 is an isolated solution of $$(W-d)z=g(z)$$, $$w=0$$. Therefore by (3.6), (3.15) and (3.16), one has

\begin{aligned} W-d&=(W-d) \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx \\ &\leq \biggl(a+b \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx \biggr)\frac{1}{\lambda_{j}^{2}} \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx+(W-d\mu_{j}) \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx \\ &=\mu_{j} \int_{\mathbb{R}^{2}}g(w_{j})w_{j}\,dx+ \int_{\mathbb{R}^{2}}h(\lambda _{j}x)w_{j}\,dx \\ &\leq\varepsilon \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx+C_{\varepsilon}\int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{p}\,dx+ \frac{1}{\lambda_{j}} \vert h \vert _{2} \vert w_{j} \vert _{2} \\ &\leq C\varepsilon+o_{n}(1), \end{aligned}

which implies a contradiction. Hence $$\{\int_{\mathbb{R}^{2}} \vert u_{j} \vert ^{2}\,dx\}$$ is bounded and then $$\{u_{j}\}$$ is bounded in E.

Secondly, for $$N=3$$, we have a simple proof. From (3.13), $$(h_{2})$$ and $$c_{\mu_{j}}\leq c_{\frac{1}{2}}$$, we get

\begin{aligned}[b] a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx &\leq a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}-3c_{\mu_{j}}+3c_{\mu_{j}} \\ &=- \int_{\mathbb{R}^{3}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx+3c_{\mu_{j}} \\ &\leq \bigl\vert \bigl(\nabla h(x),x\bigr) \bigr\vert _{2} \vert u_{j} \vert _{2}+3c_{\frac{1}{2}} \\ &\leq C \vert u_{j} \vert _{2}+3c_{\frac{1}{2}}. \end{aligned}
(3.19)

We prove directly $$\{\int_{\mathbb{R}^{3}}u_{j}^{2}\,dx\}$$ is bounded. Similar to (3.19), we obtain

\begin{aligned}[b] \frac{b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2} &\leq a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}-3c_{\mu_{j}}+3c_{\mu_{j}} \\ &\leq C \vert u_{j} \vert _{2}+3c_{\frac{1}{2}}. \end{aligned}
(3.20)

By the Hölder inequality, we have

\begin{aligned} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{3}\leq C \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{3}{4}}+C. \end{aligned}
(3.21)

By (3.3) and (3.5), for all $$t\in\mathbb{R}$$, one has

$$\bigl\vert g(t)t \bigr\vert \leq\frac{W-d}{2} \vert t \vert ^{2}+C \vert t \vert ^{6}.$$
(3.22)

From (3.11), (3.21), (3.22), $$\mu_{j}\leq 1$$ and $$D^{1,2}(\mathbb{R}^{3})\hookrightarrow L^{6}(\mathbb{R}^{3})$$, it follows that

\begin{aligned} (W-d) \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \leq& a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+b \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+(W-\mu_{j}d) \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \\ =&\mu_{j} \int_{\mathbb{R}^{3}}g(u_{j})u_{j}\,dx+ \int_{\mathbb{R}^{3}}hu_{j}\,dx \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \int_{\mathbb {R}^{3}}u_{j}^{6}\,dx+ \vert h \vert _{2} \biggl( \int_{\mathbb {R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \biggl( \int_{\mathbb {R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{3}+ \vert h \vert _{2} \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \biggl( \int_{\mathbb {R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{3}{4}} +C+ \vert h \vert _{2} \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}}, \end{aligned}

which implies that $$\{\int_{\mathbb{R}^{3}}u_{j}^{2}\,dx\}$$ is bounded. Combining with (3.19), we get that $$\{u_{j}\}$$ is bounded in E. □

### Proof of the second solution of Theorem 1.2

By $$I_{\mu_{j}}(u_{j})=c_{\mu_{j}}$$, $$I'_{\mu_{j}}(u_{j})=0$$, $$\mu_{j}\rightarrow 1^{-}$$ and Remark 3.2, we get $$I(u_{j})\rightarrow c_{1}$$ and $$I'(u_{j})\rightarrow0$$ as $$n\rightarrow +\infty$$. By Lemmas 3.5 and 3.8, there exists $$v_{0}\in E$$ such that $$u_{j}\rightarrow v_{0}$$ in E as $$n\rightarrow+\infty$$ and then $$I(v_{0})=c_{1}>0$$, $$I'(v_{0})=0$$. Be similar to (3.8), we get $$v_{0}\geq0$$. By the strong maximum principle, one has $$v_{0}>0$$. □

## 4 Conclusions

The goal of this paper is to study the multiplicity of positive solutions for the following nonlinear Kirchhoff type equation:

\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}

where a, b, V are positive constants, $$N=2$$ or 3. Under very weak conditions on f, we get that the equation has two positive solutions by using variational methods.

## References

1. Jin, J, Wu, X: Infinitely many radial solutions for Kirchhoff-type problems in $$\mathbb{R}^{N}$$. J. Math. Anal. Appl. 369, 564-574 (2010)

2. Li, H, Liao, J: Existence and multiplicity of solutions for a superlinear Kirchhoff-type equations with critical Sobolev exponent in $$\mathbb{R}^{N}$$. Comput. Math. Appl. 72, 2900-2907 (2016)

3. Li, Q, Wu, X: A new result on high energy solutions for Schrödinger-Kirchhoff type equations in $$\mathbb{R}^{N}$$. Appl. Math. Lett. 30, 24-27 (2014)

4. Liu, W, He, X: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473-487 (2012)

5. Wu, X: Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $$\mathbb{R}^{N}$$. Nonlinear Anal., Real World Appl. 12, 1278-1287 (2011)

6. Ye, Y, Tang, C: Multiple solutions for Kirchhoff-type equations in $$\mathbb{R}^{N}$$. J. Math. Phys. 54, 081508 (2013)

7. Guo, Z: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884-2902 (2015)

8. Li, G, Ye, H: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $$\mathbb{R}^{3}$$. J. Differ. Equ. 257, 566-600 (2014)

9. Liu, Z, Guo, S: Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal. 120, 1-13 (2015)

10. Tang, X, Chen, S: Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials. Calc. Var. Partial Differ. Equ. 56, 110 (2017)

11. Ekeland, I: On the variational principle. J. Math. Anal. Appl. 47, 324-353 (1974)

12. Struwe, M: Variational Methods, 2nd edn. Springer, New York (1996)

13. Jeanjean, L: On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on $$\mathbb{R}^{N}$$. Proc. R. Soc. Edinb., Sect. A, Math. 129, 787-809 (1999)

14. Jeanjean, L, Tanaka, K: A positive solution for a nonlinear Schrödinger equation on $$\mathbb{R}^{N}$$. Indiana Univ. Math. J. 54, 443-464 (2005)

15. Willem, M: Minimax Theorems. Birkhäuser, Boston (1996)

16. Li, Y, Li, F, Shi, J: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253, 2285-2294 (2012)

### Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

Not applicable.

## Funding

This work was supported by the Natural Science Foundation of Education of Guizhou Province (No. KY[2016]103, KY[2016]281, KY[2017]297); the Science and Technology Foundation of Guizhou Province (No. LH[2015]7595, LH[2016]7054).

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Each of the authors contributed to each part of this study equally. All authors read and approved the final vision of the manuscript.

### Corresponding author

Correspondence to Yu Duan.

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She, Lb., Sun, X. & Duan, Y. Multiple positive solutions for a class of Kirchhoff type equations in $$\mathbb{R}^{N}$$. Bound Value Probl 2018, 10 (2018). https://doi.org/10.1186/s13661-018-0928-8

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• DOI: https://doi.org/10.1186/s13661-018-0928-8

• 35J60
• 35B09
• 35A15

### Keywords

• Kirchhoff type equation
• variational methods
• Pohozaev equality
• monotonicity trick