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

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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.

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 , 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 , Li and Ye , Liu and Guo , Tang and Chen  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 , 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 $$N=3$$, $$(\nabla h(x),x)\in L^{2}(\mathbb{R}^{3})$$;

$$(h_{3})$$ :

h is radially symmetric.

By using Ekeland’s variational principle  and Struwe’s monotonicity trick , 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  and , 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.

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.

Proof of the main results

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

Lemma 3.1

(see  or )

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 , 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 ), 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 , 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 , 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 , 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 $$v_{0}>0$$. □

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.

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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. KY103, KY281, KY297); the Science and Technology Foundation of Guizhou Province (No. LH7595, LH7054).

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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.

Correspondence to Yu Duan.

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