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- Open Access
Multiple positive solutions for a class of Kirchhoff type equations in \(\mathbb{R}^{N}\)
- Lian-bing She1,
- Xin Sun2 and
- Yu Duan2Email author
- Received: 14 October 2017
- Accepted: 5 January 2018
- Published: 18 January 2018
Abstract
In this paper, we study the following nonlinear Kirchhoff type equation: 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.
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}$$
Keywords
- Kirchhoff type equation
- variational methods
- Pohozaev equality
- monotonicity trick
MSC
- 35J60
- 35B09
- 35A15
1 Introduction and main results
We consider the following nonlinear Kirchhoff type equation: where a, b, V are positive constants, \(N=2\) or 3.
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}$$
(1.1)
In recent years, the existence or multiplicity of solutions for the following Kirchhoff type equation where a, b are positive constants, \(N=1,2,3\), has been widely investigated by many authors, for example [1–6], 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.
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+V(x)u=f(x,u),\quad x\in\mathbb{R}^{N}, \end{aligned}$$
Very recently, Guo [7], Li and Ye [8], Liu and Guo [9], Tang and Chen [10] studied respectively the following equation: 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.
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+V(x)u=f(u),\quad x\in\mathbb{R}^{3}, \end{aligned}$$
Inspired by [7–10], 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 [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: where \(W=V+l>0\) and \(k(t)=f(t)+lt\in C(\mathbb{R}^{+},\mathbb{R}^{+})\) satisfies
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Wu=k(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}$$
(1.2)
- \((k_{1})\) :
-
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 is of class \(C^{1}\), where \(K(t)=\int_{0}^{t}k(s)\,ds\) and 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.
$$\begin{aligned} I(u)=\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx+ \frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}u^{2}\,dx- \int_{\mathbb {R}^{N}}K(u)\,dx- \int_{\mathbb{R}^{N}}hu\,dx \end{aligned}$$
$$\begin{aligned} \bigl\langle I'(u),v\bigr\rangle =& \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \int_{\mathbb{R}^{N}}(\nabla u,\nabla v)\,dx+W \int _{\mathbb{R}^{N}}uv\,dx- \int_{\mathbb{R}^{N}}k(u)v\,dx \\ &{}- \int_{\mathbb{R}^{N}}hv\,dx, \end{aligned}$$
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
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
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
where
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
$$\Phi_{\mu}(u)=A(u)-\mu B(u),\quad\forall\mu\in J, $$
$$c_{\mu}=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]} \Phi_{\mu}\bigl(\gamma(t)\bigr)>\max\bigl\{ \Phi_{\mu}(v_{1}), \Phi_{\mu}(v_{2})\bigr\} , $$
$$\Gamma=\bigl\{ \gamma\in C\bigl([0,1],X\bigr):\gamma(0)=v_{1}, \gamma(1)=v_{2}\bigr\} . $$
- (1)
\(\{u_{n}\}\) is bounded in X,
- (2)
\(\Phi_{\mu}(u_{n})\rightarrow c_{\mu}\),
- (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 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 \(\Vert u \Vert \rightarrow+\infty\).
$$\begin{aligned}& A(u)=\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx+ \frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}u^{2}\,dx- \int_{\mathbb{R}^{N}}hu\,dx, \\& B(u)= \int_{\mathbb{R}^{N}}K(u)\,dx. \end{aligned}$$
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 By (3.1), the Hölder inequality and the Sobolev inequality, for all \(\mu\in J\) and \(u\in E\), one has 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 \(\vert h\vert_{2}< m_{0}\).
$$\begin{aligned} \bigl\vert K(t) \bigr\vert \leq\frac{W+d}{4} \vert t \vert ^{2}+C \vert t \vert ^{p}. \end{aligned}$$
(3.1)
$$\begin{aligned} I_{\mu}(u) \geq&\frac{a}{2} \int_{\mathbb{R}^{2}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{2}}u^{2}\,dx- \int_{\mathbb {R}^{2}}K(u)\,dx- \int_{\mathbb{R}^{2}}hu\,dx \\ \geq&\frac{a}{2} \int_{\mathbb{R}^{2}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{2}}u^{2}\,dx-\frac{W+d}{4} \int_{\mathbb {R}^{2}}u^{2}\,dx-C \int_{\mathbb{R}^{2}} \vert u \vert ^{p}\,dx- \vert h \vert _{2} \vert u \vert _{2} \\ \geq&\frac{\min\{2a,W-d\}}{4} \Vert u \Vert ^{2}-C_{1} \Vert u \Vert ^{p}-C_{2} \vert h \vert _{2} \Vert u \Vert \\ =& \Vert u \Vert \biggl(\frac{\min\{2a,W-d\}}{4} \Vert u \Vert -C_{1} \Vert u \Vert ^{p-1}-C_{2} \vert h \vert _{2} \biggr). \end{aligned}$$
Next when \(N=3\), it follows from \((k_{1})\) and \((k_{2})\) that, for all \(t\in\mathbb{R}\), we have By (3.2), the Hölder inequality and the Sobolev inequality, for all \(\mu\in J\) and \(u\in E\), one has 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 \(\vert h\vert_{2}< m_{0}\). □
$$\begin{aligned} \bigl\vert K(t) \bigr\vert \leq\frac{W+d}{4} \vert t \vert ^{2}+C \vert t \vert ^{6}. \end{aligned}$$
(3.2)
$$\begin{aligned} I_{\mu}(u) \geq&\frac{a}{2} \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{3}}u^{2}\,dx- \int_{\mathbb {R}^{3}}K(u)\,dx- \int_{\mathbb{R}^{3}}hu\,dx \\ \geq&\frac{a}{2} \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx+ \frac{W}{2} \int_{\mathbb{R}^{3}}u^{2}\,dx-\frac{W+d}{4} \int_{\mathbb {R}^{3}}u^{2}\,dx-C \int_{\mathbb{R}^{3}} \vert u \vert ^{6}\,dx- \vert h \vert _{2} \vert u \vert _{2} \\ \geq&\frac{\min\{2a,W-d\}}{4} \Vert u \Vert ^{2}-C_{3} \Vert u \Vert ^{6}-C_{4} \vert h \vert _{2} \Vert u \Vert \\ =& \Vert u \Vert \biggl(\frac{\min\{2a,W-d\}}{4} \Vert u \Vert -C_{3} \Vert u \Vert ^{5}-C_{4} \vert h \vert _{2} \biggr). \end{aligned}$$
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 and then Hence for \(t>0\) small enough. Then we get \(c=\inf\{I(u): \Vert u \Vert \leq\rho\}<0\). \(c>-\infty\) is obvious. □
$$\begin{aligned} \int_{\mathbb{R}^{N}}\bigl(h^{2}-h\phi\bigr)\,dx\leq \int_{\mathbb {R}^{N}} \bigl\vert h^{2}-h\phi \bigr\vert \,dx \leq \vert h-\phi \vert _{2} \vert h \vert _{2}< \varepsilon \vert h \vert _{2}, \end{aligned}$$
$$\begin{aligned} \int_{\mathbb{R}^{N}}h\phi \,dx\geq \vert h \vert _{2}^{2}- \varepsilon \vert h \vert _{2}=\frac{ \vert h \vert _{2}^{2}}{2}>0. \end{aligned}$$
$$\begin{aligned} I(t\phi)\leq\frac{at^{2}}{2} \int_{\mathbb{R}^{N}} \vert \nabla\phi \vert ^{2}\,dx+ \frac{bt^{4}}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla\phi \vert ^{2}\,dx \biggr)^{2}+\frac{Wt^{2}}{2} \int_{\mathbb {R}^{N}}\phi^{2}\,dx-t \int_{\mathbb{R}^{N}}h\phi \,dx< 0 \end{aligned}$$
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 and when \(N=2\), when \(N=3\),
$$\begin{aligned} \lim_{t\rightarrow0^{+}}\frac{g(t)}{t}=0, \end{aligned}$$
(3.3)
$$\begin{aligned} \lim_{t\rightarrow+\infty}\frac{g(t)}{t^{p-1}}=0, \end{aligned}$$
(3.4)
$$\begin{aligned} \lim_{t\rightarrow+\infty}\frac{g(t)}{t^{5}}=0. \end{aligned}$$
(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 Then, by (3.6) and the Hölder inequality, one has Similarly, we can obtain that By (3.3) and (3.5), for any \(\varepsilon>0\), we have Hence, by (3.7) and the Hölder inequality, one has Similarly, we can obtain that Hence when \(N=2\) or 3, one has It is clear that and Note that Therefore we get that \(\Vert u_{n}-u \Vert \rightarrow0\) as \(n\rightarrow\infty\). □
$$\begin{aligned} \bigl\vert g(t) \bigr\vert \leq\varepsilon \vert t \vert +C_{\varepsilon} \vert t \vert ^{p-1},\quad \forall t\geq0. \end{aligned}$$
(3.6)
$$\begin{aligned}& \biggl\vert \int_{\mathbb{R}^{2}}g(u_{n}) (u_{n}-u)\,dx \biggr\vert \\& \quad\leq\varepsilon \int_{\mathbb{R}^{2}} \vert u_{n} \vert \vert u_{n}-u \vert \,dx+C_{\varepsilon}\int_{\mathbb{R}^{2}} \vert u_{n} \vert ^{p-1} \vert u_{n}-u \vert \,dx \\& \quad\leq\varepsilon \vert u_{n} \vert _{2} \vert u_{n}-u \vert _{2}+C_{\varepsilon}\biggl( \int_{\mathbb{R}^{2}} \vert u_{n} \vert ^{p}\,dx \biggr)^{\frac{p-1}{p}} \vert u_{n}-u \vert _{p} \\& \quad\leq C\varepsilon+o_{n}(1). \end{aligned}$$
$$\begin{aligned} \biggl\vert \int_{\mathbb{R}^{2}}g(u) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}$$
$$\begin{aligned} \bigl\vert g(t) \bigr\vert \leq\varepsilon\bigl( \vert t \vert + \vert t \vert ^{5}\bigr)+C_{\varepsilon} \vert t \vert ^{3},\quad \forall t\geq0. \end{aligned}$$
(3.7)
$$\begin{aligned}& \biggl\vert \int_{\mathbb{R}^{3}}g(u_{n}) (u_{n}-u)\,dx \biggr\vert \\& \quad\leq\varepsilon \int_{\mathbb{R}^{3}} \vert u_{n} \vert \vert u_{n}-u \vert \,dx+\varepsilon \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{5} \vert u_{n}-u \vert \,dx+C_{\varepsilon}\int _{\mathbb{R}^{3}} \vert u_{n} \vert ^{3} \vert u_{n}-u \vert \,dx \\& \quad\leq\varepsilon \vert u_{n} \vert _{2} \vert u_{n}-u \vert _{2}+\varepsilon \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{6}\,dx \biggr)^{\frac{5}{6}} \vert u_{n}-u \vert _{6} +C_{\varepsilon}\biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{\frac {9}{2}}\,dx \biggr)^{\frac{2}{3}} \vert u_{n}-u \vert _{3} \\& \quad\leq C\varepsilon+o_{n}(1). \end{aligned}$$
$$\begin{aligned} \biggl\vert \int_{\mathbb{R}^{3}}g(u) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}$$
$$\begin{aligned} \biggl\vert \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx \biggr\vert =o_{n}(1). \end{aligned}$$
$$\bigl\langle I'_{\mu}(u_{n})-I'_{\mu}(u),u_{n}-u \bigr\rangle =o_{n}(1) $$
$$b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u) \bigr)\,dx=o_{n}(1). $$
$$\begin{aligned} \bigl\langle I_{\mu}'(u_{n})-I'_{\mu}(u),u_{n}-u \bigr\rangle =& \biggl(a+b \int_{\mathbb {R}^{N}} \vert \nabla u_{n} \vert ^{2}\,dx \biggr) \int_{\mathbb {R}^{N}} \bigl\vert \nabla(u_{n}-u) \bigr\vert ^{2}\,dx \\ &{}+(W-\mu d) \int_{\mathbb{R}^{N}} \vert u_{n}-u \vert ^{2}\,dx \\ &{}-b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u)\bigr)\,dx \\ &{}-\mu \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx \\ \geq& \min\{a,W-\mu d\} \Vert u_{n}-u \Vert ^{2} \\ &{}-b \biggl( \int_{\mathbb{R}^{N}}\bigl( \vert \nabla u \vert ^{2}- \vert \nabla u_{n} \vert ^{2}\bigr)\,dx \biggr) \int_{\mathbb{R}^{N}}\bigl(\nabla u,\nabla(u_{n}-u)\bigr)\,dx \\ &{}-\mu \int_{\mathbb{R}^{N}} \bigl(g(u_{n})-g(u) \bigr) (u_{n}-u)\,dx. \end{aligned}$$
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 which implies that \(u_{0}^{-}=0\) and then \(u_{0}\geq0\). By the strong maximum principle, we get \(u_{0}>0\). □
$$ \begin{aligned}[b] 0&=\bigl\langle I'(u_{0}),u_{0}^{-} \bigr\rangle \\ &=-a \int_{\mathbb{R}^{N}} \bigl\vert \nabla u_{0}^{-} \bigr\vert ^{2}\,dx-b \int _{\mathbb{R}^{N}} \vert \nabla u_{0} \vert ^{2}\,dx \int_{\mathbb {R}^{N}} \bigl\vert \nabla u_{0}^{-} \bigr\vert ^{2}\,dx-W \int_{\mathbb{R}^{N}} \bigl\vert u_{0}^{-} \bigr\vert ^{2}\,dx \\ &\quad{}- \int_{\mathbb{R}^{N}}hu_{0}^{-}dx, \end{aligned} $$
(3.8)
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 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 By direct computation, we have and, by (3.9), Therefore 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 \((*)\).
$$ K(t)\geq Lt^{2}-C_{L}. $$
(3.9)
$$\begin{aligned} \operatorname{supp}w_{t}=\bigl\{ t^{2}y: y\in\operatorname{supp}w\bigr\} . \end{aligned}$$
$$\begin{aligned}& \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx=t^{2N-2} \int _{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx, \\& \int_{\mathbb{R}^{N}}w_{t}^{2}\,dx=t^{2N+2} \int_{\mathbb{R}^{N}}w^{2}\,dx \end{aligned}$$
$$\begin{aligned} \int_{\mathbb{R}^{N}}K(w_{t})\,dx =& \int_{\operatorname{supp}w_{t}}K(w_{t})\,dx \\ \geq& L \int_{\operatorname{supp}w_{t}}w_{t}^{2}\,dx-C_{L} \int_{\operatorname{supp}w_{t}}\,dx \\ \geq& Lt^{2N+2} \int_{\operatorname{supp}w}w^{2}\,dx-C_{L} \int_{\{t^{2}y:y\in B_{1}\} }\,dx \\ =&Lt^{2N+2} \int_{\mathbb{R}^{N}}w^{2}\,dx-C_{L}Ct^{2N}. \end{aligned}$$
$$\begin{aligned}& I_{\mu}(w_{t}) \\& \quad =\frac{a}{2} \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx+\frac{b}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla w_{t} \vert ^{2}\,dx \biggr)^{2}+\frac{W}{2} \int_{\mathbb{R}^{N}}w_{t}^{2}\,dx\\& \qquad {}-\mu \int _{R^{N}}K(w_{t})\,dx- \int_{\mathbb{R}^{N}}hw_{t}\,dx \\& \quad \leq\frac{at^{2N-2}}{2} \int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx+ \frac{bt^{4N-4}}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla w \vert ^{2}\,dx \biggr)^{2}+\frac{Wt^{2N+2}}{2} \int_{\mathbb {R}^{N}}w^{2}\,dx \\& \qquad {}-\frac{Lt^{2N+2}}{2} \int_{\mathbb{R}^{N}}w^{2}\,dx+ C_{L}Ct^{2N} \end{aligned}$$
By Lemma 3.3 and the definition of \(c_{\mu}\), for all \(\mu\in J\), we have Therefore, by \(I_{\mu}(0)=0\) and \(I_{\mu}(v_{2})<0\), we obtain the proof of \((**)\). □
$$0< \alpha\leq c_{1}\leq c_{\mu}\leq c_{\frac{1}{2}}\leq\max _{t\in [0,1]}I_{\frac{1}{2}}(tv_{2})< +\infty. $$
So far we have verified all the conditions of Lemma 3.1. Then there exists \(\{\mu_{j}\}\subset J\) such that
- (i)
\(\mu_{j}\rightarrow1^{-}\) as \(j\rightarrow\infty\), \(\{u_{n}^{j}\}\) is bounded in E;
- (ii)
\(I_{\mu_{j}}(u_{n}^{j})\rightarrow c_{\mu_{j}}\) as \(n\rightarrow \infty\);
- (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 Be similar to (3.8), by \(I'_{\mu_{j}}(u_{j})=0\), we obtain \(u_{j}\geq0\).
$$ a \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b(4-N)}{4} \biggl( \int_{\mathbb{R}^{N}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+ \int_{\mathbb{R}^{N}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx=Nc_{\mu_{j}}. $$
(3.13)
Firstly, we consider \(N=2\). From (3.13) and \(c_{\mu_{j}}\leq c_{\frac{1}{2}}\), we get 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 and 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 For any \(v\in C_{0}^{\infty}(\mathbb{R}^{2})\), one has and by the Lebesgue dominated convergence theorem, we have 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 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.
$$ \begin{aligned}[b] a \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx &\leq a \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{2} \biggl( \int_{\mathbb{R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2} \\ &\quad{}-2c_{\mu_{j}}+2c_{\mu_{j}} \\ &=- \int_{\mathbb{R}^{2}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx+2c_{\mu_{j}}. \end{aligned} $$
(3.14)
$$ \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx= \int_{\mathbb {R}^{2}} \vert \nabla u_{j} \vert ^{2}\,dx\leq C $$
$$ \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx= \frac{1}{\lambda _{j}^{2}} \int_{\mathbb{R}^{2}} \vert u_{j} \vert ^{2}\,dx=1. $$
(3.15)
$$ - \biggl(a+b \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx \biggr)\frac{1}{\lambda_{j}^{2}}\triangle w_{j}+(W-d \mu_{j})w_{j}=\mu_{j}g(w_{j})+h( \lambda_{j}x). $$
(3.16)
$$ \biggl\vert \int_{\mathbb{R}^{2}}h(\lambda_{j}x)v\,dx \biggr\vert \leq \vert v \vert _{2} \biggl( \int_{\mathbb{R}^{2}} \bigl\vert h(\lambda _{j}x) \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}} =\frac{1}{\lambda_{j}} \vert v \vert _{2} \vert h \vert _{2}\rightarrow0 $$
(3.17)
$$ \biggl\vert \int_{\mathbb{R}^{2}}g(w_{j})v\,dx- \int_{\mathbb{R}^{2}}g(w)v\,dx \biggr\vert \leq C \int_{\operatorname{supp}v} \bigl\vert g(w_{j})-g(w) \bigr\vert \,dx \rightarrow0. $$
(3.18)
$$ \begin{aligned} W-d&=(W-d) \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx \\ &\leq \biggl(a+b \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx \biggr)\frac{1}{\lambda_{j}^{2}} \int_{\mathbb{R}^{2}} \vert \nabla w_{j} \vert ^{2}\,dx+(W-d\mu_{j}) \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx \\ &=\mu_{j} \int_{\mathbb{R}^{2}}g(w_{j})w_{j}\,dx+ \int_{\mathbb{R}^{2}}h(\lambda _{j}x)w_{j}\,dx \\ &\leq\varepsilon \int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{2}\,dx+C_{\varepsilon}\int_{\mathbb{R}^{2}} \vert w_{j} \vert ^{p}\,dx+ \frac{1}{\lambda_{j}} \vert h \vert _{2} \vert w_{j} \vert _{2} \\ &\leq C\varepsilon+o_{n}(1), \end{aligned} $$
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 We prove directly \(\{\int_{\mathbb{R}^{3}}u_{j}^{2}\,dx\}\) is bounded. Similar to (3.19), we obtain By the Hölder inequality, we have By (3.3) and (3.5), for all \(t\in\mathbb{R}\), one has 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 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. □
$$ \begin{aligned}[b] a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx &\leq a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}-3c_{\mu_{j}}+3c_{\mu_{j}} \\ &=- \int_{\mathbb{R}^{3}}\bigl(\nabla h(x),x\bigr)u_{j}\,dx+3c_{\mu_{j}} \\ &\leq \bigl\vert \bigl(\nabla h(x),x\bigr) \bigr\vert _{2} \vert u_{j} \vert _{2}+3c_{\frac{1}{2}} \\ &\leq C \vert u_{j} \vert _{2}+3c_{\frac{1}{2}}. \end{aligned} $$
(3.19)
$$ \begin{aligned}[b] \frac{b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2} &\leq a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+\frac {b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}-3c_{\mu_{j}}+3c_{\mu_{j}} \\ &\leq C \vert u_{j} \vert _{2}+3c_{\frac{1}{2}}. \end{aligned} $$
(3.20)
$$\begin{aligned} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{3}\leq C \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{3}{4}}+C. \end{aligned}$$
(3.21)
$$ \bigl\vert g(t)t \bigr\vert \leq\frac{W-d}{2} \vert t \vert ^{2}+C \vert t \vert ^{6}. $$
(3.22)
$$\begin{aligned} (W-d) \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \leq& a \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx+b \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{2}+(W-\mu_{j}d) \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \\ =&\mu_{j} \int_{\mathbb{R}^{3}}g(u_{j})u_{j}\,dx+ \int_{\mathbb{R}^{3}}hu_{j}\,dx \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \int_{\mathbb {R}^{3}}u_{j}^{6}\,dx+ \vert h \vert _{2} \biggl( \int_{\mathbb {R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \biggl( \int_{\mathbb {R}^{3}} \vert \nabla u_{j} \vert ^{2}\,dx \biggr)^{3}+ \vert h \vert _{2} \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq&\frac{W-d}{2} \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx+C \biggl( \int_{\mathbb {R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{3}{4}} +C+ \vert h \vert _{2} \biggl( \int_{\mathbb{R}^{3}}u_{j}^{2}\,dx \biggr)^{\frac{1}{2}}, \end{aligned}$$
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: 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.
$$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}$$
Declarations
Acknowledgements
The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.
Availability of data and materials
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).
Authors’ contributions
Each of the authors contributed to each part of this study equally. All authors read and approved the final vision of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics and Information Engineering, Liupanshui Normal College, LiuPanshui, Guizhou, 553004, People’s Republic of China
(2)
Collect of Science, GuiZhou University of Engineering Science, BiJie, Guizhou, 551700, People’s Republic of China
References
- Jin, J, Wu, X: Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 369, 564-574 (2010) MathSciNetView ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
- Liu, W, He, X: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 39, 473-487 (2012) MathSciNetView ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
- Ye, Y, Tang, C: Multiple solutions for Kirchhoff-type equations in \(\mathbb{R}^{N}\). J. Math. Phys. 54, 081508 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Guo, Z: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884-2902 (2015) MathSciNetView ArticleMATHGoogle Scholar
- 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) View ArticleMATHGoogle Scholar
- Liu, Z, Guo, S: Existence of positive ground state solutions for Kirchhoff type problems. Nonlinear Anal. 120, 1-13 (2015) MathSciNetView ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
- Ekeland, I: On the variational principle. J. Math. Anal. Appl. 47, 324-353 (1974) MathSciNetView ArticleMATHGoogle Scholar
- Struwe, M: Variational Methods, 2nd edn. Springer, New York (1996) View ArticleMATHGoogle Scholar
- 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) View ArticleMATHGoogle Scholar
- Jeanjean, L, Tanaka, K: A positive solution for a nonlinear Schrödinger equation on \(\mathbb{R}^{N}\). Indiana Univ. Math. J. 54, 443-464 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Willem, M: Minimax Theorems. Birkhäuser, Boston (1996) View ArticleMATHGoogle Scholar
- 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) MathSciNetView ArticleMATHGoogle Scholar
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