Skip to content

Advertisement

  • Research
  • Open Access

Existence of nontrivial solution for a nonlocal problem with subcritical nonlinearity

Advances in Difference Equations20182018:359

https://doi.org/10.1186/s13662-018-1823-4

  • Received: 4 August 2018
  • Accepted: 28 September 2018
  • Published:

Abstract

In this paper, we consider the following new nonlocal Dirichlet boundary value problem:
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=\lambda u+g(x,u),& x\in \Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$
(0.1)
where a and b are positive, λ is a positive parameter, \(0\leq\lambda< a\lambda_{1}\), \(\lambda_{1}\) is the first eigenvalue of operator −Δ. Under appropriate assumptions on the function g which is of subcritical growth, we obtain a nontrivial solution.

Keywords

  • Nonlocal problem
  • Nontrivial solution
  • Subcritical nonlinearity

MSC

  • 35B33
  • 35B38
  • 35B09

1 Introduction and main result

In this paper, we consider the following new nonlocal Dirichlet boundary value problem:
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=\lambda u+g(x,u),& x\in \Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$
(1.1)
where a and b are positive, λ is a positive parameter.

The search for a nontrivial solution of problem (1.1) is a new subject and of great significance. We put forward a new nonlocal term \(a-b\int _{\Omega}|\nabla u|^{2}\,dx\), which is different from the well known nonlocal term \(a+b\int_{\Omega}|\nabla u|^{2}\,dx\) and presents a lot of interesting difficulties.

Recently, mathematical studies have focused on the existence of solutions of the Kirchhoff type problem
$$ \textstyle\begin{cases} -(a+b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=g(x,u),& x\in\Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$
where \(a>0\), \(b>0\) and Ω is either a smooth bounded domain in \(\mathbb{R}^{N}\) or \(\Omega=\mathbb{R}^{N}\). The results about problem with subcritical nonlinearity can be seen in [15] and the critical cases in [613]. Here we do not present the results in detail, someone who is interested in them can consult the references therein.
However, there are only few results about problem (1.1). When \(\lambda =0\) and \(g(x,u)=|u|^{p-2}u\) was of subcritical growth, Yin and Liu [14] considered
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u= \vert u \vert ^{p-2}u,& x\in\Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$
and obtained existence and multiplicity of nontrivial solutions. When \(\lambda=0\) and \(g(x,u)=f_{\lambda}(x)|u|^{p-2}u\), Lei [15] considered
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=f_{\lambda}(x) \vert u \vert ^{p-2}u,& x\in\Omega, \\ u=0,& x\in\partial\Omega. \end{cases} $$
Under some special conditions and for \(1< p<2\), the author obtained two solutions. Lei [16] also investigated
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u=\frac{\lambda}{u^{\gamma }},& x\in\Omega, \\ u=0,& x\in\partial\Omega, \end{cases} $$
and, when \(0<\gamma<1\) and \(0<\lambda<\lambda_{*}\), at least two positive solutions were obtained. Wang [17] studied a nonlocal problem involving critical exponent, namely
$$ \textstyle\begin{cases} -(a-b\int_{\Omega} \vert \nabla u \vert ^{2}\,dx)\Delta u= \vert u \vert ^{2}u+\mu f(x),& x\in \mathbb{R}^{4}, \\ u\in D^{1, 2}(\mathbb{R}^{4}), \end{cases} $$
for which infinitely many positive solutions and at least two positive solutions were found for \(\mu=0\) and \(\mu\in(0, \mu_{*}]\). For some other important results the interested reader is also referred to [1821].
We are inspired by the above articles and consider a new problem which is different from the mentioned above. Assume that nonlinearity g satisfies the following assumptions:
\((g_{1})\)

g is continuous, \(1\leq i\leq N\), \(|g(x, u)|\leq C(1+|u|^{p-1})\) for some \(C>0\) and \(2< p<2^{*}\), where \(2^{*}=\frac {2N}{N-2}\) if \(N\geq3\), \(2^{*}=\infty\) if \(N=1\text{ or }2\);

\((g_{2})\)

\(g(x, u)=o(u)\) uniformly in x as \(u\rightarrow0\);

\((g_{3})\)

\(u\mapsto\frac{g(x, u)}{u}\) is positive for \(u\neq 0\), nonincreasing on \((-\infty, 0)\) and nondecreasing on \((0, +\infty)\).

Now, we state our main result.

Theorem 1.1

Suppose that conditions \((g_{1})\)\((g_{3})\) and \(0\leq\lambda< a\lambda _{1}\) hold, then problem (1.1) has a nontrivial solution.

2 Preliminary results

In this section, we present the variational results which will be used in the proof of Theorem 1.1. Let \(E:=H_{0}^{1}(\Omega)\) be endowed with the usual norm
$$\Vert u \Vert =\langle u, u\rangle^{1/2}= \biggl( \int_{\Omega} \vert \nabla u \vert ^{2} \biggr)^{1/2}. $$
The usual norm in the Lebesgue space \(L^{p}(\Omega)\) is denoted by \(|u|_{p}\).
A function \(u\in E\) is called a weak solution of problem (1.1) if
$$a \int_{\Omega}\nabla u \nabla v\,dx-b \Vert u \Vert ^{2} \int_{\Omega}\nabla u \nabla v\,dx=\lambda \int_{\Omega}uv\,dx- \int_{\Omega}g(x, u)v\,dx,\quad \forall v\in E. $$
Moreover, our assumptions imply that the solutions of (1.1) are the critical points of the functional defined in E by
$$I(u)=\frac{a}{2} \Vert u \Vert ^{2}-\frac{b}{4} \Vert u \Vert ^{4}-\frac{\lambda}{2} \int _{\Omega} \vert u \vert ^{2}\,dx- \int_{\Omega}G(x, u)\,dx. $$
It is easy to see for \(\forall u, v\in E\),
$$\bigl\langle I'(u), v\bigr\rangle =a \int_{\Omega}\nabla u \nabla v\,dx-b \Vert u \Vert ^{2} \int_{\Omega}\nabla u \nabla v\,dx-\lambda \int_{\Omega}uv\,dx- \int _{\Omega}g(x, u)v\,dx. $$
Let \(\lambda_{i}\) (\(i=1,2,\dots\)) be the eigenvalues of operator −Δ with zero Dirichlet boundary condition. It is well known that each eigenvalue \(\lambda_{i}\) is positive, isolated and has finite multiplicity, the smallest eigenvalue \(\lambda_{1}\) being simple and \(\lambda_{i}\rightarrow\infty\) as \(i\rightarrow\infty\). Here we only need the first eigenvalue of −Δ, where \(\lambda_{1}=\inf_{u\neq 0}\frac{\int_{\Omega}|\nabla u|^{2}}{\int_{\Omega}|u|^{2}}\) and assume that \(0\leq\lambda< a\lambda_{1}\).

3 Proof of Theorem 1.1

In this section, we will prove Theorem 1.1, so from now on we always suppose that \((g_{1})\)\((g_{3})\) hold. First, \((g_{1})\) and \((g_{2})\) imply that for each \(\varepsilon>0\) there is a \(C_{\varepsilon}>0\) such that
$$ \bigl\vert g(x, u) \bigr\vert \leq\varepsilon \vert u \vert +C_{\varepsilon} \vert u \vert ^{p-1} \quad\text{for all } u \in\mathbb{R}. $$
(3.1)
And using \((g_{2})\) and \((g_{3})\), one can easily check that
$$ G(x, u)\geq0 \quad\text{and} \quad g(x, u)u\geq2G(x, u)>0 \quad \text{if } u\neq0. $$
(3.2)

Lemma 3.1

If \(0\leq\lambda< a\lambda_{1}\), then there exists a sequence \(\{u_{n}\} \subset E\) satisfying \(I(u_{n})\rightarrow c\), \(I'(u_{n})\rightarrow 0\), where \(0< c<\frac{a^{2}}{4b}\).

Proof

For \(\lambda_{1}=\inf_{u\neq0}\frac{\int_{\Omega}|\nabla u|^{2}}{\int_{\Omega}|u|^{2}}\), then
$$\biggl(a-\frac{\lambda}{\lambda_{1}} \biggr) \int_{\Omega} \vert \nabla u \vert ^{2}\leq a \int_{\Omega} \vert \nabla u \vert ^{2}-\lambda \int_{\Omega} \vert u \vert ^{2}\leq a \int_{\Omega} \vert \nabla u \vert ^{2}. $$
Also by (3.1), we can choose a sufficiently small \(\varepsilon=\frac {\lambda_{1}}{2} (a-\frac{\lambda}{\lambda_{1}} )\), and then
$$ \begin{aligned} I(u)&=\frac{a}{2} \Vert u \Vert ^{2}-\frac{b}{4} \Vert u \Vert ^{4}- \frac{\lambda}{2} \int _{\Omega} \vert u \vert ^{2}- \int_{\Omega}G(x, u) \\ &\geq\frac{1}{2} \biggl(a-\frac{\lambda}{\lambda_{1}} \biggr) \int_{\Omega } \vert \nabla u \vert ^{2}- \frac{b}{4}\biggl( \int_{\Omega} \vert \nabla u \vert ^{2} \biggr)^{2} -\frac{\varepsilon}{2} \int_{\Omega} \vert u \vert ^{2}-\frac{C_{\varepsilon}}{p} \int _{\Omega} \vert u \vert ^{p} \\ &\geq\frac{1}{2} \biggl(a-\frac{\lambda}{\lambda_{1}} \biggr) \int_{\Omega } \vert \nabla u \vert ^{2}- \frac{b}{4} \Vert u \Vert ^{4}-\frac{\varepsilon}{2\lambda _{1}} \int_{\Omega} \vert \nabla u \vert ^{2}- \frac{C_{1}C_{\varepsilon}}{p} \Vert u \Vert ^{p} \\ &\geq\frac{1}{4} \biggl(a-\frac{\lambda}{\lambda_{1}} \biggr) \Vert u \Vert ^{2}-\frac{b}{4} \Vert u \Vert ^{4}- \frac{C_{1}C_{\varepsilon}}{p} \Vert u \Vert ^{p}, \end{aligned} $$
Since \(4< p<2^{*}\), for small enough \(\rho>0\), for all \(u\in E\) and \(\| u\|=\rho\), it holds that \(I(u)=\gamma>0\). On the other hand, for \(u\neq 0\) and \(t\in\mathbb{R}\),
$$ I(tu)=\frac{at^{2}}{2} \Vert u \Vert ^{2}-\frac{bt^{4}}{4} \Vert u \Vert ^{4}-\frac{\lambda t^{2}}{2} \int_{\Omega} \vert u \vert ^{2}- \int_{\Omega}G(x, tu), $$
so that when \(t\rightarrow\infty\), we have \(I(tu)\rightarrow-\infty\). This means that there is a \(t_{1}\) such that \(u_{1}=t_{1}u\in E\), \(\| u_{1}\|>\rho\) and \(I(u_{1})<0\). As a consequence, by the mountain pass lemma without (PS) condition [22], there exists a sequence \(\{ u_{n}\}\subset E\) such that \(I(u_{n})\rightarrow c\), \(I'(u_{n})\rightarrow0\) for
$$c=\inf_{h\in\Gamma}\max_{u\in h([0, 1])}I(u)\geq\gamma>0, $$
where
$$\Gamma=\bigl\{ h\in C\bigl([0, 1], E\bigr): h(0)=0, h(1)=u_{1} \bigr\} . $$
Because
$$ \begin{aligned} \max_{t\in[0, 1]}I(tu_{1})&= \max_{t\in[0, 1]} \biggl\{ \frac{at^{2}}{2} \Vert u_{1} \Vert ^{2}-\frac{bt^{4}}{4} \Vert u_{1} \Vert ^{4}-\frac{\lambda t^{2}}{2} \int _{\Omega} \vert u_{1} \vert ^{2}- \int_{\Omega}G(x, tu_{1}) \biggr\} \\ &< \max_{t\in[0, 1]} \biggl\{ \frac{at^{2}}{2} \Vert u_{1} \Vert ^{2}-\frac {bt^{4}}{4} \Vert u_{1} \Vert ^{4} \biggr\} \\ &\leq\frac{a^{2}}{4b}, \end{aligned} $$
it is easy to obtain that \(0< c<\frac{a^{2}}{4b}\) according to the definition of c. □

Lemma 3.2

Under the condition \(c<\frac{a^{2}}{4b}\), I satisfies the \((PS)_{c}\) condition, i.e., any \((PS)_{c}\) sequence of I has a convergent subsequence.

Proof

We drew on the experience of [14]. Let \(\{u_{n}\} \subset E\) be such that \(I(u_{n})\rightarrow c\), \(I'(u_{n})\rightarrow 0\). Since by (3.2)
$$\begin{aligned} c+o(1)&=I(u_{n})-\frac{1}{2}\bigl\langle I'(u_{n}), u_{n}\bigr\rangle \\ &=\frac{a}{2} \Vert u_{n} \Vert ^{2}- \frac{b}{4} \Vert u_{n} \Vert ^{4}- \frac{\lambda }{2} \int_{\Omega} \vert u_{n} \vert ^{2}- \int_{\Omega}G(x, u_{n}) \\ &\quad{}- \biggl[\frac{a}{2} \Vert u_{n} \Vert ^{2}- \frac{b}{2} \Vert u_{n} \Vert ^{4}- \frac{\lambda }{2} \int_{\Omega} \vert u_{n} \vert ^{2}- \frac{1}{2}g(x, u_{n}) \biggr] \\ &\geq\frac{b}{4} \Vert u_{n} \Vert ^{4}, \end{aligned}$$
we know that \(\{u_{n}\}\) is bounded in E. By passing to a subsequence, still denoted \(\{u_{n}\}\), we may assume that there is a \(u\in E\) such that
$$ \begin{aligned} &u_{n}\rightharpoonup u \quad \text{in } E, \\ &u_{n}\rightarrow u \quad \text{in } L^{s}(\Omega) \text{ for } s\in [1, 2^{*}), \\ &u_{n}(x)\rightarrow u(x) \quad \text{for a.e. } x\in\Omega. \end{aligned} $$
On account of
$$ \begin{aligned}o(1)&=\bigl\langle I'(u_{n}), u_{n}-u\bigr\rangle \\ &=\bigl(a-b \Vert u_{n} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{n} \nabla(u_{n}-u)- \lambda \int_{\Omega}u_{n}(u_{n}-u)- \int_{\Omega}g(x, u_{n}) (u_{n}-u) \end{aligned} $$
and
$$\biggl\vert \int_{\Omega}u_{n}(u_{n}-u) \biggr\vert \leq \biggl( \int_{\Omega } \vert u_{n} \vert ^{2} \biggr)^{\frac{1}{2}} \biggl( \int_{\Omega} \vert u_{n}-u \vert ^{2} \biggr)^{\frac{1}{2}}, $$
also by (3.1)
$$ \begin{aligned} & \biggl\vert \int_{\Omega}g(x, u_{n}) (u_{n}-u) \biggr\vert \\ &\quad \leq\varepsilon \biggl\vert \int_{\Omega}u_{n}(u_{n}-u) \biggr\vert +C_{\varepsilon } \biggl\vert \int_{\Omega} \vert u_{n} \vert ^{p-2}u_{n}(u_{n}-u) \biggr\vert \\ &\quad \leq\varepsilon \biggl( \int_{\Omega} \vert u_{n} \vert ^{2} \biggr)^{\frac{1}{2}} \biggl( \int_{\Omega} \vert u_{n}-u \vert ^{2} \biggr)^{\frac{1}{2}} +C_{\varepsilon} \biggl( \int_{\Omega}\bigl( \vert u_{n} \vert ^{p-1} \bigr)^{\frac{p}{p-1}} \biggr)^{\frac{p-1}{p}} \biggl( \int_{\Omega}\bigl( \vert u_{n}-u \vert ^{p}\bigr) \biggr)^{\frac{1}{p}}, \end{aligned} $$
because \(u_{n}\rightarrow u\) in \(L^{s}(\Omega)\), \(s\in[1, 2^{*})\), the above two formulas show that when \(n\rightarrow\infty\),
$$ \bigl(a-b \Vert u_{n} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{n} \nabla(u_{n}-u) \rightarrow0. $$
(3.3)
If there exists a subsequence of \(\{u_{n}\}\), still denoted \(\{u_{n}\} \), satisfying \(\|u_{n}\|^{2}\rightarrow\frac{a}{b}\), define a functional
$$\varphi(u)=\frac{\lambda}{2} \int_{\Omega} \vert u \vert ^{2}+ \int_{\Omega}G(x, u), \quad u\in E. $$
Then
$$\bigl\langle \varphi'(u), v\bigr\rangle =\lambda \int_{\Omega}uv+ \int_{\Omega}g(x, u)v, \quad u, v\in E, $$
and
$$\bigl\langle \varphi'(u_{n})-\varphi'(u), v\bigr\rangle =\lambda \int_{\Omega }(u_{n}-u)v+ \int_{\Omega}\bigl[g(x, u_{n})-g(x, u)\bigr]v. $$

Claim. \(\langle\varphi'(u_{n})-\varphi'(u), v\rangle \rightarrow0\), \(\forall v\in E\).

Firstly,
$$\lambda \int_{\Omega}(u_{n}-u)v\leq\lambda \biggl( \int_{\Omega } \vert u_{n}-u \vert ^{2} \biggr)^{\frac{1}{2}} \biggl( \int_{\Omega} \vert v \vert ^{2} \biggr)^{\frac{1}{2}}, $$
since \(u_{n}\rightarrow u\) in \(L^{2}(\Omega)\), thus \(\lambda\int_{\Omega }(u_{n}-u)v\rightarrow0\).
Secondly, to prove the claim, we only need to prove
$$ \lim_{n\rightarrow\infty} \int_{\Omega} \bigl\vert g(x, u_{n})-g(x, u) \bigr\vert \vert v \vert =0. $$
(3.4)
If (3.4) is not true, then there exist a constant \(\varepsilon_{0}>0\) and a subsequence \(u_{k_{i}}\) such that
$$ \int_{\Omega} \bigl\vert g(x, u_{k_{i}})-g(x, u) \bigr\vert \vert v \vert \geq\varepsilon_{0}, \quad \forall i\in \mathbb{N}, $$
(3.5)
Since \(u_{n}\rightarrow u\) in \(L^{p}(\Omega)\), passing to a subsequence if necessary, we can assume that \(\sum_{i=1}^{\infty }|u_{k_{i}}-u|_{p}^{p}<+\infty\). Set
$$\omega(x)= \Biggl[\sum_{i=1}^{\infty} \bigl\vert u_{k_{i}}(x)-u(x) \bigr\vert ^{p} \Biggr]^{\frac {1}{p}},\quad \forall x\in\Omega. $$
Then \(\omega\in L^{p}(\Omega)\). Note that for \(\forall i\in\mathbb{N}\), \(x\in\Omega\),
$$ \begin{aligned}[b] & \bigl\vert g(x, u_{k_{i}})-g(x, u) \bigr\vert \vert v \vert \\ &\quad \leq\bigl( \bigl\vert g(x, u_{k_{i}}) \bigr\vert + \bigl\vert g(x, u) \bigr\vert \bigr) \vert v \vert \\ &\quad \leq\bigl[\varepsilon\bigl( \vert u_{k_{i}} \vert + \vert u \vert \bigr)+C_{\varepsilon }\bigl( \vert u_{k_{i}} \vert ^{p-1}+ \vert u \vert ^{p-1}\bigr)\bigr] \vert v \vert \\ &\quad \leq\bigl[2^{2}\varepsilon\bigl( \vert u_{k_{i}}-u \vert + \vert u \vert \bigr)+2^{p}C_{\varepsilon }\bigl( \vert u_{k_{i}}-u \vert ^{p-1}+ \vert u \vert ^{p-1}\bigr) \bigr] \vert v \vert \\ &\quad \leq\bigl[2^{2}\varepsilon\bigl( \vert \omega \vert + \vert u \vert \bigr)+2^{p}C_{\varepsilon}\bigl( \vert \omega \vert ^{p-1}+ \vert u \vert ^{p-1}\bigr)\bigr] \vert v \vert \\ &\quad :=f(x), \end{aligned} $$
(3.6)
and
$$ \begin{aligned} \int_{\Omega}f(x)\,dx&= \int_{\Omega}\bigl[2^{2}\varepsilon\bigl( \vert \omega \vert + \vert u \vert \bigr)+2^{p}C_{\varepsilon}\bigl( \vert \omega \vert ^{p-1}+ \vert u \vert ^{p-1}\bigr)\bigr] \vert v \vert \\ &\leq2^{2}\varepsilon\bigl( \vert \omega \vert _{2}+ \vert u \vert _{2}\bigr) \vert v \vert _{2}+2^{p}C_{\varepsilon } \bigl( \vert \omega \vert _{p}^{p-1}+ \vert u \vert _{p}^{p-1}\bigr) \vert v \vert _{p}< +\infty. \end{aligned} $$
(3.7)
Since \(u_{k_{i}}\rightarrow u\) a.e. in Ω, then by (3.6), (3.7) and Lebesgue Dominated Convergence Theorem, we have
$$\lim_{i\rightarrow\infty} \int_{\Omega} \bigl\vert g\bigl(x, u_{k_{i}}(x)\bigr)-g \bigl(x, u(x)\bigr) \bigr\vert \vert v \vert =0, $$
which contradicts (3.5). Hence (3.4) holds. Then the claim follows. By arbitrariness of v, then
$$\bigl\Vert \varphi'(u_{n})-\varphi'(u) \bigr\Vert _{E'}\rightarrow0, $$
and \(\varphi'(u_{n})\rightarrow\varphi'(u)\) in \(E'\). While \(\langle I'(u_{n}), v\rangle=(a-b\|u_{n}\|^{2})\langle u_{n}, v\rangle-\langle \varphi'(u_{n}), v\rangle\), \(\langle I'(u_{n}), v\rangle\rightarrow0\), \(a-b\|u_{n}\|^{2}\rightarrow 0\), hence \(\varphi'(u_{n})\rightarrow0\), i.e.,
$$\bigl\langle \varphi'(u), v\bigr\rangle =\lambda \int_{\Omega}uv+ \int_{\Omega}g(x, u)v=0,\quad\forall v\in E, $$
and then we have
$$\lambda u(x)+g\bigl(x, u(x)\bigr)=0\quad\text{for a.e. }x\in\Omega, $$
by the fundamental lemma of the variational method (see [23]). It follows that \(u=0\). So
$$\varphi(u_{n})=\frac{\lambda}{2} \int_{\Omega} \vert u_{n} \vert ^{2}+ \int_{\Omega }G(x, u_{n})\rightarrow\frac{\lambda}{2} \int_{\Omega} \vert u \vert ^{2}+ \int _{\Omega}G(x, u)=0. $$
Hence we see that \(I(u_{n})=\frac{a}{2}\|u_{n}\|^{2}-\frac{b}{4}\| u_{n}\|^{4}-\frac{\lambda}{2}\int_{\Omega}|u_{n}|^{2}-\int_{\Omega }G(x, u_{n})\rightarrow\frac{a^{2}}{4b}\) from \(\|u_{n}\|^{2}\rightarrow \frac{a}{b}\). This is a contradiction to \(I(u_{n})\rightarrow c<\frac {a^{2}}{4b}\). Then \(a-b\|u_{n}\|^{2}\rightarrow0\) is not true and any subsequence of \(\{a-b\|u_{n}\|^{2}\rightarrow0\}\) does not converge to zero. Therefore there exists a \(\delta>0\) such that \(|a-b\|u_{n}\| ^{2}|>\delta>0\) when n is large enough. It is clear that \(\{a-b\| u_{n}\|^{2}\rightarrow0\}\) is bounded. It follows from (3.3) that \(\int_{\Omega}\nabla u_{n}\nabla (u_{n}-u)\rightarrow0\). So \(\|u_{n}\|\rightarrow\|u\|\). Hence \(u_{n}\rightarrow u\) in E due to the uniform convexity of E. □

Proof of Theorem 1.1

According to Lemma 3.1, there exists a sequence \(\{u_{n}\}\in E\) satisfying \(I(u_{n})\rightarrow c>0\), \(I'(u_{n})\rightarrow0\). By Lemma 3.2, \(\{u_{n}\}\), which is the sequence obtained by Lemma 3.1, possesses a convergent to u subsequence (still denoted by \(\{u_{n}\} \)). So it follows from the continuity that \(I(u_{n})\rightarrow c>0\), \(I'(u_{n})\rightarrow0\). But \(I(0)=0\), therefore \(u\neq0\), that is, u is a nontrivial solution of problem (1.1). □

Declarations

Acknowledgements

We would like to thank the referee for his/her valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper. The authors are supported by The Inner Mongolia Autonomous Region university scientific research project (NJZY18021) and Postdoctoral research project of Inner Mongolia University (21100-5175504) and Inner Mongolia Normal University introduces high-level scientific research projects (2016YJRC005) and Research project of Inner Mongolia Normal University (2016ZRYB001).

Availability of data and materials

Not applicable.

Funding

Not applicable.

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final 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)
Mathematics Sciences College, Inner Mongolia Normal University, Hohhot, P.R. China
(2)
College of Mathematics, Changchun Normal University, Changchun, P.R. China

References

  1. Batkam, C.-J.: An elliptic equation under the effect of two nonlocal terms. Math. Methods Appl. Sci. 39, 1535–1547 (2016) MathSciNetView ArticleGoogle Scholar
  2. Delgado, M., Figueiredo, G.-M., Gayte, I., Morales-Rodrigo, C.: An optimal control problem for a Kirchhoff-type equation. ESAIM Control Optim. Calc. Var. 23, 773–790 (2017) MathSciNetView ArticleGoogle Scholar
  3. Ding, L., Meng, Y.J., Xiao, S.W., Zhang, J.L.: Existence of two positive solutions for indefinite Kirchhoff equations in \(\mathbb{R}^{3}\). Electron. J. Differ. Equ. 2016, 35 (2016) View ArticleGoogle Scholar
  4. Mao, A.M., Zhu, X.C.: Existence and multiplicity results for Kirchhoff problems. Mediterr. J. Math. 14, 58 (2017) MathSciNetView ArticleGoogle Scholar
  5. Tang, X.H., Cheng, B.T.: Ground state sign-changing solutions for Kirchhoff type problems in bounded domains. J. Differ. Equ. 261, 2384–2402 (2016) MathSciNetView ArticleGoogle Scholar
  6. He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb{R}^{3}\). J. Differ. Equ. 252, 1813–1834 (2012) View ArticleGoogle Scholar
  7. Liu, Z., Guo, S.J., Fang, Y.Q.: Positive solutions of Kirchhoff type elliptic equations in \(\mathbb{R}^{4}\) with critical growth. Math. Nachr. 290, 367–381 (2016) View ArticleGoogle Scholar
  8. Peng, C.Q.: The existence and concentration of ground-state solutions for a class of Kirchhoff type problems in \(\mathbb{R}^{3}\) involving critical Sobolev exponents. Bound. Value Probl. 64, 64 (2017) View ArticleGoogle Scholar
  9. Wu, X.: Existence of nontrivial solutions and hingh energy solutions for Schrödinger–Kirchhoff-type equations in \(\mathbb{R}^{N}\). Nonlinear Anal. 12, 1278–1287 (2011) View ArticleGoogle Scholar
  10. Yang, L., Liu, Z.S., Ouyang, Z.: Multiplicity results for the Kirchhoff type equations with critical growth. Appl. Math. Lett. 63, 118–123 (2017) MathSciNetView ArticleGoogle Scholar
  11. Zhang, J.: The Kirchhoff type Schrödinger problem with critical growth. Nonlinear Anal., Real World Appl. 28, 153–170 (2016) MathSciNetView ArticleGoogle Scholar
  12. Zhang, J.: The critical Neumann problem of Kirchhoff type. Appl. Math. Comput. 274, 519–530 (2016) MathSciNetGoogle Scholar
  13. Zhang, C.H., Liu, Z.S.: Multiplicity of nontrivial solutions for a critical degenerate Kirchhoff type problem. Appl. Math. Lett. 69, 87–93 (2017) MathSciNetView ArticleGoogle Scholar
  14. Yin, G.S., Liu, J.S.: Existence and multiplicity of nontrivial solutions for a nonlocal problem. Bound. Value Probl. 2015, 26 (2015) MathSciNetView ArticleGoogle Scholar
  15. Lei, C.Y., Liao, J.F., Suo, H.M.: Multiple positive solutions for a class of nonlocal problems involving a sign-changing potential. Electron. J. Differ. Equ. 9, 1 (2017) Google Scholar
  16. Lei, C.Y., Chu, C.M., Suo, H.M.: Positive solutions for a nonlocal problem with singularity. Electron. J. Differ. Equ. 85, 1 (2017) MathSciNetMATHGoogle Scholar
  17. Wang, Y., Suo, H.M., Lei, C.Y.: Multiple positive solutions for a nonlocal problem involving critical exponent. Electron. J. Differ. Equ. 275, 1 (2017) MathSciNetMATHGoogle Scholar
  18. Liang, S., Shi, S.: Soliton solutions to Kirchhoff type problems involving the critical growth in \(\mathbb{R}^{N}\). Nonlinear Anal. 81, 31–41 (2013) MathSciNetView ArticleGoogle Scholar
  19. Liang, S., Zhang, J.: Existence of solutions for Kirchhoff type problems with critical nonlinearity in \(\mathbb{R}^{3}\). Nonlinear Anal., Real World Appl. 17, 126–136 (2014) MathSciNetView ArticleGoogle Scholar
  20. Liang, S., Zhang, J.: Multiplicity of solutions for the noncooperative Schrödinger–Kirchhoff system involving the fractional p-Laplacian in \(\mathbb{R}^{N}\). Z. Angew. Math. Phys. 68, 63 (2017) View ArticleGoogle Scholar
  21. Zhang, B.: Multi-peak solutions for a nonlinear Schrödinger–Poisson system including critical growth in \(\mathbb{R}^{3}\). Adv. Differ. Equ. 2016, 189 (2016) View ArticleGoogle Scholar
  22. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14, 349–381 (1973) MathSciNetView ArticleGoogle Scholar
  23. Lu, W.D.: The Variational Method in Differential Equation. Sichuan University Press, Sichuan (1995) Google Scholar

Copyright

© The Author(s) 2018

Advertisement