Ground States of Nonlinear Schrödinger-Kirchhoff Type Equation

Weidan Li

College of Science, University of Shanghai for Science and Technology, Shanghai, China.

DOI: 10.4236/jamp.2020.82025                

Abstract

In this paper, we considered with the following Schr?dinger-Kirchhoff type problem: -(a+b ∫_R^N|∇u|²dx)Δu + V(x)u = f(x,u) in R^N. We put forward general assumptions on the nonlinearity f with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with a Nehari-Pankov manifold by using a linking theorem.

Keywords

Schrödinger-Kirchhoff, Linking, Ceramisequence, Fatou’s Lemma

1. Introduction

Consider the following nonlinear Schrödinger-Kirchhoff type problem:

$-\left(a+b{\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\right)\Delta u+V\left(x\right)u=f\left(x,u\right)$ in $R^N$ (1.1)

where constants $a>0,b\ge 0$， $V\in \left(R^N,R\right)$ and $f\in \left(R^N\times R,R\right)$ satisfy some assumptions.

In (1.1), if $a=1,b=0$, then Equation (1.1) is the following well-known Schrödinger equation:

$-\Delta u+V\left(x\right)u=f\left(x,u\right)$, in $R^N$ (1.2)

The Schrödinger equation has been studied by Brezis and Lieb [1] for a general class of autonomous, and by many authors [2] [3] [4] for periodic data.

If $V\left(x\right)\equiv 0$ and $R^N$ are replaced by a smooth bounded domain $\Omega \in R^N$, then Equation (1.1) is a Dirichlet problem of Kirchhoff type [5] :

$\{\begin{array}{l}-\left(a+b{\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\right)\Delta u=f\left(x,u\right),\text{on}\Omega \\ u=0,\text{in}\partial \text{Ω}\end{array}$ (1.3)

Many interesting studies by variational methods can be found in [6] - [11]. It is related to the following stationary analogue of the equation:

$u_{tt}-\left(a+b{\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\right)\Delta u=f\left(x,u\right)$ (1.4)

which is an extension of classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early studies of Kirchhoff equation can be refer to in [12] - [19] and the references therein.

In order to reduce some extra statement we need to describe the eigenvalue of the Schrodinger operator $-a\Delta +V$. We are considered the sequence $\left\{{\lambda }_i\right\}$, ${\lambda }_i<0<{\lambda }_{i+1}<{\lambda }_{\infty }=\underset{i\to \infty }{\lim }{\lambda }_i$ of minimax values, it is known that ${\lambda }_{\infty }$, if finite, is lower bound of the essential spectrum $\sigma \left(-a\Delta +V\right)$ of Schrodinger operator $-a\Delta +V$ (many details can see [20]).

Our aim is to study ground state solutions to (1.1) with a class of nonlinearities. The energy functional $I:X\to R$ given by:

$I\left(u\right)=\frac{a}{2}{\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}+\frac{b}{4}{\left({\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\right)}^2+\frac{1}{2}{\displaystyle {\int }_{R^N}V\left(x\right)u^2\text{d}x}-{\displaystyle {\int }_{R^N}F\left(x,u\right)\text{d}x}$ (1.5)

Set $F\left(x,u\right)={\displaystyle {\int }_0^{u}f\left(x,s\right)\text{d}s}$. We make the following assumptions:

(V) $V\in C\left(R^N\right)$ bounded from below, there is $M>0$, such that

$\text{meas}\left\{x\in R^N|V\left(x\right)<M\right\}<\infty$

(F) $f:R^N\times R\to R$ is measurable in $x\in R^N$ and continuous in $u\in R$ for a.e. $x\in R^N$.

$F:R^N\times R\to R$ is differentiable with respect to the second variable $u\in R$ and $F\left(x,0\right)=0$ for a.e. $x\in R^N$.

(F1) There are $A>0$ and $2<p<2^{*}=\frac{2N}{N-2}$, such that

$\left|f\left(x,u\right)\right|\le A\left(1+{\left|u\right|}^{p-1}\right)$ for all $u\in R$ and a.e. $x\in R^N$.

(F2) $f\left(x,u\right)=o\left(\left|u\right|\right)$ as $\left|u\right|\to 0$ uniformly in $x\in R^N$.

Our goal is to find a ground state solution of Equation (1.1) i.e., a critical point being a minimizer of I on the Nehari-Pankov manifold defined as follows:

$\mathcal{N}=\left\{u\in X\backslash X^{-}|\langle I^{\prime }\left(u\right),u\rangle =0\text{and}\langle I^{\prime }\left(u\right),v\rangle =0\text{for}\text{any}v\in X^{-}\right\}$.

Since $\mathcal{N}$ contains all critical points of I, then a ground state is a energy solution. To conquer the linking geometry of I and to structure $\mathcal{N}$, we give the following assumptions:

(F3) $f\left(x,u\right)u\ge 4F\left(x,u\right)\ge 0$ for all $u\in R$ and a.e. $x\in R^N$.

(F4) $\frac{F\left(x,u\right)}{{\left|u\right|}^4}\to \infty$ as $\left|u\right|\to \infty$ uniformly in $x\in R^N$.

We have our main result as follow.

Theorem 1.1. Suppose that (V), (F), (F1)-(F4) are satisfied. Then (1.1) has a ground state solution, there is a nontrivial critical point of I, such that $I\left(u\right)=\underset{N}{\inf }I$.

Our approach is the same a new linking-type result of [21] involving the Nehari-Pankov manifold. In the next section, we present a critical point theory and variational setting. In Section 3, we state some relevant lemmas and prove Theorem 1.1.

2. Preliminaries and Variational Setting

Set Hilbertspace

$H^1\left(R^N\right)=\left\{u\in L^2\left(R^N\right),\nabla u\in L^2\left(R^N\right)\right\}$

with the norm

${\Vert u\Vert }_{H^1}={\left({\displaystyle {\int }_{R^N}\left({\left|\nabla u\right|}^2+u^2\right)\text{d}x}\right)}^{\frac{1}{2}}$

To prove our theorems, let

$X=\left\{u\in H^1\left(R^N\right)|{\displaystyle {\int }_{R^N}\left(a{\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\text{d}x}<\infty \right\}$

and define the inner product of X by the following formula:

$\langle u,v\rangle ={\displaystyle {\int }_{R^N}a\langle \nabla u^{+},\nabla u^{+}\rangle +V\left(x\right)\langle u^{+},u^{+}\rangle \text{d}x}-{\displaystyle {\int }_{R^N}a\langle \nabla u^{-},\nabla u^{-}\rangle +V\left(x\right)\langle u^{-},u^{-}\rangle \text{d}x}$

and norm given by $\Vert u\Vert ={\langle u,u\rangle }^{\frac{1}{2}}$, where $u=u^{+}+u^{-}$, $v=v^{+}+v^{-}\in X=X^{+}\oplus X^{-}$,

therefore X is a Hilbert space with the norm $\Vert \Vert$. It is easy to see that

${\displaystyle {\int }_{R^N}\left(a{\left|\nabla u\right|}^2+V\left(x\right)u^2\right)\text{d}x}={\Vert u^{+}\Vert }^2-{\Vert u^{-}\Vert }^2$.

Let $X^{-}$ is the finite dimentional space spanned by the negative eigenfunctions with ${\lambda }_i<0,\left(i=1,\cdots ,n\right)$ and let $X^{+}={\left(X^{-}\right)}^{\perp }$. In view of (V), we may find continuous projections $u^{+}$ and $u^{-}$ of X onto $X^{+}$ and $X^{-}$, respectively, such that $X=X^{+}+X^{-}$ and $X^{+}$ is the positive eigenspace and $X^{-}$ is the negative eigenspace of the operator $-a\Delta +V$.

For any $t\in \left[2,2^{*}\right]$, the embedding $x\to L^t\left(R^N\right)$ is continuous. Consequently, there is a constant ${\gamma }_t>0$, such that

${\Vert u\Vert }_t\le {\gamma }_t\Vert u\Vert ,\forall u\in X$ (2.1)

Moreover, we know that under assumption (V), the embedding $x\to L^t\left(R^N\right)$ is compact for any $t\in \left[2,2^{*}\right)$ by lemma 3.4 in [22].

Then the relative functional of (1.1) can be writed by

$I\left(u\right)=\frac{1}{2}{\Vert u^{+}\Vert }^2-\frac{1}{2}{\Vert u^{-}\Vert }^2+\frac{1}{4}{\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}F\left(x,u\right)\text{d}x}$

and under assumptions (F1) and (F2), $I\in C^1\left(X,R\right)$, for any $u,v\in X$,

$\langle I^{\prime }\left(u\right),v\rangle =\left(a+b{\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\right){\displaystyle {\int }_{R^N}\nabla u\nabla v\text{d}x}+{\displaystyle {\int }_{R^N}V\left(x\right)uv\text{d}x}-{\displaystyle {\int }_{R^N}f\left(x,u\right)v\text{d}x}$.

From the assumptions that (F1) and (F2), we know a functional $I\in C^1\left(X,R\right)$, then the following conditions hold:

(A1) I is lower semicontinuous.

(A2) $I^{\prime }$ is weak continuous.

Let $P\subset X\backslash X^{-}$ and $P\ne \varnothing$. The linking geometry of I is described by the following conditions:

(A3) There exists $r>0$, such that $\underset{u\in X^{+},\Vert u\Vert =r}{\inf }I\left(u\right)>0$.

(A4) For every $u\in P$, there exists $R\left(u\right)>r$, such that $\underset{\partial M\left(u\right)}{\sup }I\le I\left(0\right)=0$ where $M\left(u\right):=\left\{tu+v\in X|t\ge 0,v\in X^{-}\text{and}\Vert tu+v\Vert \le R\left(u\right)\right\}$.

(A5) If $u\in N$, then $I\left(u\right)\ge I\left(tu+v\right)$ for any $t\ge 0$ and $v\in X^{-}$ where $N=\left\{u\in P|\langle I^{\prime }\left(u\right),u\rangle =0\text{and}\langle I^{\prime }\left(u\right),v\rangle =0\text{for}\text{any}v\in X^{-}\right\}$.

If $P=X\backslash X^{-}$, then N has been introduced by Pankov [23].

For any $A\subset X,I\subset \left[0,\infty \right)$, such that $0\in I$, and $h:A\times I\to X$, we collect the following assumptions [24] :

(h1) h is a continuous.

(h2) $h\left(0,u\right)=u$ for all $u\in A$.

(h3) $I\left(u\right)\ge I\left(h\left(u,t\right)\right)$ for all $\left(u,t\right)\in A\times I$.

(h4) each $\left(u,t\right)\in A\times I$ has an open neighborhood W in the product topology of $\left(X,T\right)$ and I such that the set $\left\{v-h\left(v,s\right):\left(v,s\right)\in W\cap \left(A\times I\right)\right\}$ is contained in a finite dimentional subspace of X.

Theorem 2.1. (linking theory [21] [24]) Suppose that $I\in C^1\left(X,R\right)$ satisfies conditions A1-A4).

Then there exists a Cerami sequence $\left\{u_n\right\}$ (i.e. $I\left(u_n\right)\to c$, $\left(1+\Vert u_n\Vert \right)I^{\prime }\left(u_n\right)\to$ ), where

$c=\underset{u\in P}{\inf }\underset{h\in \Gamma \left(u\right)}{\inf }\underset{u^{\prime }\in M\left(u\right)}{\sup }I\left(h\left(u^{\prime },1\right)\right)>0$

$\Gamma \left(u\right)=\left\{h\in C\left(M\left(u\right)\times \left[0,1\right]\right)|h\text{satisfies}\left(\text{h1}\right)\text{-}\left(\text{h4}\right)\right\}$.

Suppose that in addition (A5) hold. Then $c\le {\inf }_{N}I$, and if $c\ge I\left(u\right)$, for some critical point $u\in P$. then

$c=\underset{N}{\inf }I$

where $N=\left\{u\in P|\langle I^{\prime }\left(u\right),u\rangle =0\text{and}\langle I^{\prime }\left(u\right),v\rangle =0\text{for}\text{any}v\in X^{-}\right\}$.

3. Proof of Theorem

We need the following lemmas.

Lemma 3.1. Assume that $I\in C^1\left(X,R\right)$ and assumptions (V), (F), (F1) and (F2) are satisfied, then conditions (A1) and (A2) are hold.

Proof. Set $J\left(u\right)={\displaystyle {\int }_{R^N}F\left(x,u\right)\text{d}x}$. According to (1.5), it suffices to show that $J\left(u\right)$ is weakly continuous on X.

For any $\epsilon >0$, by (F1) and (F2), there is $C_{\epsilon }>0$, such that

$\left|f\left(x,u\right)\right|=\epsilon \left|u\right|+C_{\epsilon }{\left|u\right|}^{p-1}$ for $u\in R$ (3.1)

let $\left\{u_n\right\}\subset X$ and $u_n\rightharpoonup u$ in X, then $\left\{u_n\right\}$ is bounded in X and converges to u in $L^t\left(R^N\right)$, where $2\le t<2^{*}$, by (3.1) we have

$\begin{array}{c}\left|J\left(u_n\right)-J\left(u\right)\right|=\left|{\displaystyle {\int }_{R^N}F\left(x,u_n\left(x\right)\right)-F\left(x,u\left(x\right)\right)\text{d}x}\right|\\ =\left|{\displaystyle {\int }_{R^N}{\displaystyle {\int }_0^{1}f\left(x,u+s\left(u_n-u\right)\right)\left(u_n-u\right)\text{d}s}\text{d}x}\right|\\ \le {\displaystyle {\int }_{R^N}\left[\epsilon \left|u+s\left(u_n-u\right)\right|+C_{\epsilon }{\left|u+s\left(u_n-u\right)\right|}^{p-1}\right]}\left|u_n-u\right|\text{d}x\\ \le {\displaystyle {\int }_{R^N}\left[\epsilon \left(\left|u\right|+\frac{1}{2}\left|u_n-u\right|\right)+C_{\epsilon ,p}\left({\left|u\right|}^{p-1}+{\left|u_n-u\right|}^{p-1}\right)\right]}\left|u_n-u\right|\text{d}x\\ \le c\left(\epsilon +C_{\epsilon ,p}{\Vert u_n-u\Vert }_p\right)\le C_{\epsilon ,n}\end{array}$

Therefore, $J\left(u\right)$ is weakly continuous on X. This shows (A1) and (A2) hold.□

Lemma 3.2. Assume that assumptions (F), (F1) and (F2) are satisfied, then condition (A3) are hold.

Proof. For any $0<\epsilon <\frac{1}{4{\gamma }_2^2}$ ( ${\gamma }_2$ is depended on (2.1)), by (3.1) we know that there is $C_{\epsilon }>0$ such that

$\left|F\left(x,u\right)\right|\le \frac{\epsilon }{2}{\left|u\right|}^2+\frac{C_{\epsilon }}{p}{\left|u\right|}^p$ for $u\in R$. (3.2)

Note that (F4) implies $p>4$, then there is $r>0$, set $S=\left\{u\in X^{+},\Vert u\Vert =r\right\}$, for every $u\in S$, from (3.2) we have

$\begin{array}{c}I\left(u\right)=\frac{1}{2}\Vert u^{+}\Vert +\frac{1}{4}{\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}F\left(x,u\right)\text{d}x}\\ \ge \frac{1}{2}{\Vert u^{+}\Vert }^2-\frac{\epsilon }{2}{\Vert u\Vert }_2^2-\frac{C_{\epsilon }}{p}{\Vert u\Vert }_p^p\\ \ge \frac{1}{4}\left(1-{\gamma }_2^2\epsilon \right){\Vert u^{+}\Vert }^2\end{array}$

Therefore

$\underset{u\in s}{\inf }I\left(u\right)\ge \frac{1}{4}\left(1-{\gamma }_2^2\epsilon \right)r^2>0$

This shows (A3) hold. □

Hence similarly as in [17] [19], we obtain that conditions (A1)-(A3) are hold. Moreover, obviously, $I\left(0\right)=0$, I has the linking geometry.

Lemma 3.3. Assume that assumptions (V), (F), (F1)-(F4) are satisfied, then condition (A4) is hold.

Proof. Choose a fixed $u\in P$ and there are $t_n>0$ and $v_n\in X^{-}$ such that $I\left(t_{n}u+v_n\right)>0$ and $\Vert t_{n}u+v_n\Vert \to \infty$ as $n\to \infty$, let $z_n=\frac{t_{n}u+v_n}{\Vert t_{n}u+v_n\Vert }$ and we let $z_n\rightharpoonup z$ in X and $z_n\left(x\right)\to z\left(x\right)$ a.e. $R^N$ for some $z\in X$. Since

$0<I\left(t_{n}u+v_n\right)\le \frac{1}{2}{\Vert t_n{u}^{+}\Vert }^2-\frac{1}{2}{\Vert t_n{u}^{-}+v_n\Vert }^2+\frac{1}{4}{\Vert t_n{u}^{+}\Vert }^4$

then

$0<\frac{t_n^4}{{\Vert t_{n}u+v_n\Vert }^4}{\Vert u^{+}\Vert }^4={\Vert z_n^{+}\Vert }^4\le {\Vert z_n\Vert }^4=1$

and we may assume that $z^{+}\ne 0$. Hence $z\ne 0$ and

$\left|t_{n}u\left(x\right)+v_n\left(x\right)\right|=\left|z_n\left(x\right)\right|\Vert t_{n}u+v_n\Vert \to \infty$ as $n\to \infty$ and $z\left(x\right)\ne 0$.

Then by (F4) and Fatou’s lemma, we have

$0<\frac{I\left(t_{n}u+v_n\right)}{{\Vert t_{n}u+v_n\Vert }^4}\le \frac{{\gamma }_2^4}{4}-{\displaystyle {\int }_{R^N}\frac{F\left(x,t_{n}u+v_n\right)}{{\left|t_{n}u+v_n\right|}^4}{\left|z_n\right|}^4\text{d}x}\to -\infty$ as $t_n\to \infty$

and is a contradiction. This shows (A4) hold. □

Remark 3.4. The inspection of proof of lemma 3.3, then due to conclusion of that lemma 3.3, for any $t\ge 0$, $v\in X^{-}$ such that (A5) holds.

Lemma 3.5. Assume that (V), (F), (F1) and (F2) are satisfied. If $\left\{u_n\right\}\in X$ is a bounded Cerami sequence in X, then $\left\{u_n\right\}$ has a convergent subsequence.

Proof: Let $\left\{u_n\right\}$ is a Cerami sequence in X, that is $I\left(u_n\right)\to c$ and $\left(1+\Vert u_n\Vert \right)I^{\prime }\left(u_n\right)\to 0$.

From $I\left(u_n\right)\to \infty$ as $\Vert u_n\Vert \to \infty$, we get that I is coercive.

Hence it is easy to get that $\left\{u_n\right\}$ is bounded. Since

$o\left(\Vert u_n\Vert \right)=\langle I^{\prime }\left(u_n\right),u_n\rangle ={\Vert u_n^{+}\Vert }^2-{\Vert u_n^{-}\Vert }^2+{\Vert u_n\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u_n\right)u_n\text{d}x}$

One has

$o\left(1\right)+{\Vert u_n^{-}\Vert }^2={\Vert u_n^{+}\Vert }^2+{\Vert u_n\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u_n\right)u_n\text{d}x}$ (3.3)

Passing to a subsequence, we may assume that $u_n\rightharpoonup u$ in X. Therefore

$0=\langle I^{\prime }\left(u\right),u\rangle ={\Vert u^{+}\Vert }^2-{\Vert u^{-}\Vert }^2+{\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u\right)u\text{d}x}$

and we have

${\Vert u^{-}\Vert }^2={\Vert u^{+}\Vert }^2+{\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u\right)u\text{d}x}$ (3.4)

Moreover, since $\dim X^{-}<\infty$, we have that $u_n^{-}\to u^{-}$. Then combine (3.3) with (3.4) push out

$\underset{n\to \infty }{\lim }\left({\Vert u_n^{+}\Vert }^2+{\Vert u_n\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u_n\right)u_n\text{d}x}\right)={\Vert u^{+}\Vert }^2+{\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u\right)u\text{d}x}$ (3.5)

It follows from (A2) and Fatou’s lemma that

$\underset{n\to \infty }{\underset{\_}{\lim }}\left({\Vert u_n\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u_n\right)u_n\text{d}x}\right)\ge {\Vert u\Vert }_2^4-{\displaystyle {\int }_{R^N}f\left(x,u\right)u\text{d}x}$ (3.6)

Combining (3.5) and (3.6), we can get

$\underset{n\to \infty }{\underset{\_}{\lim }}{\Vert u_n^{+}\Vert }^2\le {\Vert u^{+}\Vert }^2$

and then $\Vert u_n^{+}\Vert \to \Vert u^{+}\Vert$. Consequently, $u_n\to u$ in X.

This shows lemma’s conclusion hold. □

Proof of Theorem 1.1. Observe that (F3) implies that $P=X\backslash X^{-}$. In view of theorem 2.1, there is a bounded Cerami sequence $\left\{u_n\right\}$ and by lemma 3.5, $\left\{u_n\right\}$ has a convergent subsequence. Then passing to a subsequence, let $u_n\rightharpoonup u$ in X, then $u_n\left(x\right)\to u\left(x\right)$ a.e. in $R^N$. From lemma 3.1, we can obtain that I is lower semicontinuous. One has that

${\displaystyle {\int }_{R^N}{\left|\nabla u\right|}^2\text{d}x}\le \underset{n\to \infty }{\underset{\_}{\lim }}{\displaystyle {\int }_{R^N}{\left|\nabla u_n\right|}^2\text{d}x}$

Observe that by (F3) and in view of Fatou’s lemma that

$c=\underset{n\to \infty }{\underset{\_}{\lim }}I\left(u_n\right)=\underset{n\to \infty }{\underset{\_}{\lim }}\left(I\left(u_n\right)-\frac{1}{4}\langle I^{\prime }\left(u_n\right),u_n\rangle \right)\ge I(u)$

Since $u\in N$, then by theorem 2.1 we have

$c=\underset{N}{\inf }I=I(u)$

This shows the main result hold. □

4. Conclusion

To sum up the above arguments, we through the linking theory to prove the existence of ground state solution of Schrödinger-Kirchhoff type equation.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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