数学杂志  2018, Vol. 38 Issue (4): 743-750   PDF    
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汪敏庆
黄文念
方立婉
一类非线性Schrödinger-Maxwell方程基态解的存在性
汪敏庆, 黄文念, 方立婉    
广西师范大学数学与统计学院, 广西 桂林 541006
摘要:本文研究了如下Schrödinger-Maxwell方程基态解的存在性问题 $ \left\{\begin{array}{ll} -\triangle u+V(x)u+K(x)\phi(x)u=b(x)|u|^{p-1}u+\lambda g(x, u)~~ {\rm in} \ \mathbb{R}^{3}, \\ -\triangle\phi=K(x)u^{2}~~ {\rm in} \ \mathbb{R}^{3}, \end{array} \right. $ 其中$\lambda>0, V(x)\in C^{1}(\mathbb{R}^{3}, \mathbb{R})$, 且$V(x)>0$.在$K, g, b$满足一定的假设条件下, 且$0<p<1$时, 利用变分法和临界点理论, 获得了基态解的存在性.该结论推广了文献[7]的结果.
关键词Schrödinger-Maxwell方程    非线性    基态解    变分法    临界点理论    Nehari流形    
EXISTENCE OF GROUND STATE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER-MAXWELL EQUATIONS
WANG Min-qing, HUANG Wen-nian, FANG Li-wan    
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China
Abstract: In this paper, we study the existence of ground state solutions for SchrödingerMaxwell equations $ \left\{\begin{array}{ll} -\triangle u+V(x)u+K(x)\phi(x)u=b(x)|u|^{p-1}u+\lambda g(x, u)~~ {\rm in} \ \mathbb{R}^{3}, \\ -\triangle\phi=K(x)u^{2}~~ {\rm in} \ \mathbb{R}^{3}, \end{array} \right. $ where $\lambda>0, V(x)\in C^{1}(\mathbb{R}^{3}, \mathbb{R})$ and $V(x)>0.$ Under certain assumptions on $K, g$ and $0<p<1$, we obtain the ground state solutions by using variational methods and critical point theory, which promotes the results of literature [7].
Key words: Schrödinger-Maxwell equations     nonlinear     ground state solution     variational methods     critical point theory     Nehari manifold    
1 引言及主要结果

考虑以下非线性Schrödinger-Maxwell方程基态解的存在性.

$ \left\{\begin{array}{ll} -\triangle u+V(x)u+K(x)\phi(x)u=b(x)|u|^{p-1}u+\lambda g(x, u)~~ {\rm in} \ \mathbb{R}^{3}, \\ -\triangle\phi=K(x)u^{2}~~ {\rm in} \ \mathbb{R}^{3}, \end{array} \right. $

这样的方程又被称为Schrödinger-Poisson方程.在量子力学中, 该方程可描述带电粒子与电磁场的相互作用(关于物理方面的更多的描述可详见文献[1]).

过去的几十年里, 在临界点理论和变分法的帮助下, 类似于系统(1.1)的系统的解的存在性、不存在性和多重性得到了广泛的研究, 具体可参考文献[2-4].进一步地, 当$V(x)\equiv K(x)\equiv1$, $f(x, u)=|u|^{p-1}u, 1<p<5$时的情形, 在文献[5]中已经得到研究.无独有偶, 在文献[6]中, Sun运用变形的喷泉定理证明了系统(1.1)无穷多解的存在性.文献[7]中, Ma和Sun运用变形的山路定理, 在特定的假设下得到了系统(1.1)基态解的存在性.更多的结论可参阅文献[8, 9, 10, 11]等.

$V, K, b, g$有以下假设

(V) $V(x)\in C^{1}(\mathbb{R}^{3}, \mathbb{R}), \inf \limits_{x\in\mathbb{R}^3}V(x)\geq a_{1}>0$, 其中$ a_{1}>0$是一个常数.对每一个$M>0, {\rm meas}\{x\in \mathbb{R}^{3}, V(x)\leq M\}<\infty$.

(K) $K\in L^\infty(\mathbb{R}^3, \mathbb{R})$, 对任意的$x\in \mathbb{R}^3$, 有$K(x)\geq 0$.

(B) $b:\mathbb{R}^{3}\rightarrow\mathbb{R}^{+}$是一个正连续函数, 并且$b\in L^{\frac{2}{1-P}}$, 其中$0<p<1$.

(g1) $g(x, u)\in C(\mathbb{R}^3\times\mathbb{R}, \mathbb{R})$, $g$$x_{i}~(i=1, 2, 3)$中是1 -周期的, 且$|g(x, u)|\leq C(1+|u|^{q-1})$, 其中$2<q<6$.

(g2)当$u\rightarrow 0$时, 对所有的$x\in \mathbb{R}^{3}$, 有$g(x, u)=o(u)$.

(g3)当$u\rightarrow \infty$时, 对所有的$x\in \mathbb{R}^{3}$, 有$\lim \limits_{u \rightarrow\infty}\frac{g(x, u)}{u^{3}}=\infty$.

(g4)对任意的$(x, u)\in (\mathbb{R}^{3}, \mathbb{R})$, 有$\frac{1}{4}g(x, u)u-G(x, u)\geq 0, $其中$G(x, u)=\displaystyle\int_{0}^{u}g(x, s)ds.$

对于系统$(1)$, 主要的结果如下

定理 1.1  假设(V), (K), (B), (g1)-(g4)成立, 则系统$(1.1)$存在一个基态解, 其中$C>0$表示一系列不同的正常数.

2 预备知识及相关引理

定义下列函数空间$H^{1}(\mathbb{R}^{3})=\{u\in L^{2}(\mathbb{R}^{3})|\nabla u\in (L^{2}(\mathbb{R}^{3})^{3}\}.$对应的范数为

$ \|u\|_{H^{1}}:=(\displaystyle\int _{\mathbb{R}^{3}}(|\nabla u|^{2}+u^{2})dx)^{\frac{1}{2}}. $

定义函数空间$ D^{1, 2}(\mathbb{R}^{3}):=\{u\in L^{2^{*}}(\mathbb{R}^{3}):\nabla u\in (L^{2}(\mathbb{R}^{3})^{3}\}. $对应范数为

$ \|u\|_{D^{1, 2}}:=(\displaystyle\int _{\mathbb{R}^{3}}(|\nabla u|^{2})dx)^{\frac{1}{2}}. $

$ E=\{u\in H^{1}(\mathbb{R}^{3}):\displaystyle\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+V(x)u^{2})dx<+\infty\}. $

$E$是一个Hilbert空间, 对应的内积为

$ (u, v)_{E}=\displaystyle\int_{\mathbb{R}^{3}}(\nabla u\nabla v+V(x)uv)dx, $

范数为$\|u\|=(u, u)^{\frac{1}{2}}$.

$\|\cdot\|$$L^{s}(\mathbb{R}^{3})$下的范数, $H=H_{r}^{1}(\mathbb{R}^{3})$$H(\mathbb{R}^{3})$空间中径向函数的子空间, 则$H$可以紧嵌入$L^{s}(\mathbb{R}^{3})$, 其中$s\in (2, 6)$[16].再记

$ S=\inf \limits_{u\in \mathcal{D}^{1, 2}(\mathbb{R}^3), |u|_6=1}|\nabla u|_2, \gamma_{s}=\sup \limits_{u\in H^{1}(\mathbb{R}^3), \|u\|=1}|u|_{s}, $

$E$是连续嵌入到$L^{s}(\mathbb{R}^{3})$中的, $s\in [2, 2^{*}]$, 这里的$2^{*}=6$是在三维空间里Sobolev嵌入的临界指数.因此, 存在一个常数$C>0$, 使得

$ \begin{equation} \|u\|_{L^{s}}\leq C\|u\|, \forall u\in E, \end{equation} $ (2.1)

其中

$ \|u\|_{L^{s}}:=(\displaystyle\int _{\mathbb{R}^{3}}|u|^{s}dx)^{\frac{1}{s}}, \forall s\in [2, 2^{*}] $

是在Lebesgue空间$L^{s}(\mathbb{R}^{3})$下的范数.

因为$K\in L^\infty(\mathbb{R}^3, \mathbb{R})$, 故对每一个$u\in E$, 由Hölder不等式, 有

$ \begin{align} \displaystyle\int_{\mathbb{R}^{3}}Ku^{2}vdx&\leq\left(\displaystyle\int_{\mathbb{R}^{3}}(Ku^{2})^{\frac{6}{5}}dx\right)^{\frac{5}{6}}\left(\displaystyle\int_{\mathbb{R}^{3}}(|v|^{6})dx\right)^{\frac{1}{6}}\nonumber\\ &\leq C\|u\|_{12/5}^{2}\|v\|_{\mathcal{D}}, \forall v\in\mathcal{D}^{1, 2}(\mathbb{R}^{3}). \end{align} $ (2.2)

由Lax-Milgram定理(详见文献[11])可知, 对任意的$u\in E$, 存在唯一的$\phi_{u}\in\mathcal{D}^{1, 2}(\mathbb{R}^3), $使得

$ -\triangle\phi_{u}=Ku^{2}. $ (2.3)

对于$\phi_{u}$, 可以写成下列积分形式

$ \phi_{u}(x)=\frac{1}{4\pi}\displaystyle\int_{\mathbb{R}^{3}}\frac{K(y)u^{2}(y)}{|x-y|}dy. $

故由$x\in\mathbb{R}^{3}, K(x)\geq 0$可知, $\phi_{u}(x)\geq 0$.结合$(2.1)$$(2.2)$式, 有

$ \|\phi_{u}\|_{\mathcal{D}}\leq C\|u\|_{12/5}^{2}. $

因此由Hölder不等式和$(2.1)$式, 有

$ \begin{align} \displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx&\leq C(\displaystyle\int_{\mathbb{R}^{3}}|\phi_{u}|^{6}dx)^{\frac{1}{6}}(\displaystyle\int_{\mathbb{R}^{3}}|u|^{\frac{12}{5}}dx)^{\frac{5}{6}}\nonumber\\ &\leq C\|\phi_{u}\|_{\mathcal{D}^{1, 2}}\|u\|_{12/5}^{2} \leq C\|u\|_{12/5}^{4}\leq c\|u\|^{4}. \end{align} $ (2.4)

定义泛函$I$: $E\rightarrow\mathbb{R}$

$ I(u, \phi)\;\;=\;\;\frac{1}{2}\displaystyle\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+V(x)u^{2})dx-\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}(|\nabla \phi|^{2})dx\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;+\frac{1}{2}\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p+1}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}G(x, u)dx. $

从上面的讨论可知$I$$C^{1}$的, 并且$I$的临界点就是问题$(1.1)$的解.进一步地, 由$(2.3)$式有

$ \Phi(u)=I(u, \phi)=\frac{1}{2}\|u\|^{2}+\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p+1}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}G(x, u)dx. $

如果$u\in E$是泛函$\Phi$的一个临界点(也就是$\Phi'(u)=0$), 则$(u, \phi_{u})$是系统$(1.1)$的一个解.进一步地, 对任意的$u, v\in E$, 有

$ \langle\Phi'(u), v\rangle=\displaystyle\int_{\mathbb{R}^{3}}(\nabla u\nabla v+V(x)uv)dx+\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}uvdx-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p-1}uvdx-\lambda\displaystyle\int_{\mathbb{R}^{3}}g(x, u)vdx. $

特别地,

$ \langle\Phi'(u), u\rangle=\|u\|^{2}+\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p-1}u^{2}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}g(x, u)udx. $

定义对应的Nehari流形为$ \mathcal{N}=\{u\in E:\langle \Phi'(u), u\rangle=0\}. $$(X, \|\cdot\|)$为Hilbert空间, $\{e_{j}\}$为其一组标准正交基, 令$X_{j}={\rm span}\{e_{j}\}, Y_{k}=\bigoplus_{j=0}^{k}X_{j}, Z_{k}=\overline{\bigoplus_{j=k}^{\infty}X_{j}}.$

定义 2.1  设$E$是一个实Banach空间, $\Phi\in C^{1}(E, \mathbb{R}), c\in \mathbb{R}$.当$n\rightarrow \infty, u_{n}\in E $时, 如果对任意满足

$ \Phi(u_{n})\rightarrow c, \Phi'(u_{n})\rightarrow 0 $

的序列$\{u_{n}\}\subset X$都有收敛的子列, 则称$\Phi$满足(PS)$_{c}$条件.

为了证明定理1.1, 我们将会利用以下形式的山路定理(详见文献[12, 13, 14]).

定理 2.2 [12, 13, 14]  设$E$是一个实Banach空间, $\Phi\in C^{1}(E, \mathbb{R}), \Phi(0)=0$, 对任意的$c>0$, $\Phi$满足满足(PS)$_{c}$条件, 且

(ⅰ) 存在$\rho, \alpha>0, $使得$\Phi|_{tial B_{\rho}}\geq\alpha;$

(ⅱ) 存在$e\in E\backslash B_{\rho}$, 使得$\Phi(e)\leq 0$.

$\Phi$有一个临界值$c\geq \alpha.$

引理 2.3 [15]  对任意的  $2\leq s< 2^{*}, $$\beta_{k}:=\sup \limits_{u\in Z_{k}, \|u\|_{E}=1}|u|_{L^{s}}\rightarrow 0, k\rightarrow \infty. $

3 定理1.1的证明

引理 3.1  若(V), (K), (B), (g1)-(g4)成立, $e\in E\backslash \{0\}$, 则

(ⅰ) 存在$\rho, \alpha>0, $使得$\Phi|_{tial B_{\rho}}\geq\alpha.$

(ⅱ) 当$|t|\rightarrow\infty, \Phi(te)=-\infty$.

(ⅰ)由假设(g1), (g2)可知, 对任意的$\varepsilon>0, $存在$C(\varepsilon)>0, $使得对所有的$x\in\mathbb{R}^{3}, u\in\mathbb{R}, $

$ |g(x, u)|\leq\varepsilon|u|+C(\varepsilon)|u|^{q-1}. $ (3.1)

因此, 由中值定理, 有

$ |G(x, u)|=|G(x, u)-G(x, 0)| =\left|\displaystyle\int_{0}^{1}g(x, su)uds\right| \leq\frac{\varepsilon}{2}|u|^{2}+\frac{C(\varepsilon)}{q}|u|^{q}. $ (3.2)

由引理$2.3$$p\in(0, 1)$可知, 存在$R_{e}>0, $使得当$\|u\|\geq R_{e}$时, 有

$ \frac{|\lambda|\gamma_{s}^{p+1}}{p+1}|b|\cdot\|u\|^{p+1}\leq\frac{1}{4}\|u\|^{2}. $ (3.3)

于是, 对于$u\in Z_{k}$, $\|u\|\geq R_{e}$, 有

$ \begin{align} \Phi(u)&=\frac{1}{2}\|u\|^{2}+\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p+1}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}G(x, u)dx\nonumber\\ &\geq\frac{1}{2}\|u\|^{2}-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p+1}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}G(x, u)dx\nonumber\\ &\geq\frac{1}{2}\|u\|^{2}-\frac{1}{4}\|u\|^{2}-\frac{\lambda\varepsilon}{2}\|u\|^{2}-\frac{\lambda C(\varepsilon)}{q}\|u\|^{q}\nonumber\\ &=(\frac{1}{4}-\frac{\lambda\varepsilon}{2})\|u\|^{2}-\frac{\lambda\varepsilon}{2}\|u\|^{2}-\frac{\lambda C(\varepsilon)}{q}\|u\|^{q}. \end{align} $ (3.4)

$\varepsilon>0$足够小, 使得$(\frac{1}{4}-\frac{\lambda\varepsilon}{2})>0, $由于$q>2$, 所以有

$ \Phi(u)\geq\frac{1}{2}(\frac{1}{4}-\frac{\lambda\varepsilon}{2})\|\rho\|^{2}\equiv\alpha>0, \forall\|u\|=\rho, $

其中$\rho=\left[\frac{q}{2\lambda C(\varepsilon)}(\frac{1}{4}-\frac{\lambda\varepsilon}{2})\right]^{\frac{1}{q-2}}$.

(ⅱ) 由(g{3})可知, 对任意的$M>0$, 存在$\xi=\xi(M)>0$, 使得对所有的$x\in\mathbb{R}^{3}, |u|>\xi, $

$ G(x, u)\geq M|u|^{4}, $ (3.5)

由(g1), (g2)可知, 存在$M_{1}=M_{1}(M)>0, $使得对所有的$x\in\mathbb{R}^{3}, 0<|u|\leq\xi, $

$ \frac{|g(x, u)u|}{|u|^{2}}\leq M_{1}. $ (3.6)

$(3.6)$式和中值定理可知, 对所有的$x\in\mathbb{R}^{3}, |u|\leq\xi, $

$ G(x, u)\leq\frac{M_{1}}{2}|u|^{2}. $ (3.7)

$\tilde{M}=M|\xi|^{2}+\frac{M_{1}}{2}$, 结合$(3.5)$$(3.7)$式, 有

$ G(x, u)\geq M|u|^{4}-\tilde{M}|u|^{2}, (x, u)\in(\mathbb{R}^{3}\times\mathbb{R}). $ (3.8)

$(3.8)$式, 有

$ \begin{align*} \Phi(te)&\leq\frac{1}{2}t^{2}\|e\|^{2}+Ct^{4}\|e\|^{4}-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)t^{p+1}|e|^{p+1}dx-\lambda Mt^{4}\|e\|_{4}^{4}+\lambda \tilde{M}t^{2}\|e\|_{2}^{2}\nonumber\\ &\leq\frac{1}{2}t^{2}\|e\|^{2}+Ct^{4}\|e\|^{4}-\lambda Mt^{4}\|e\|_{4}^{4}+\lambda \tilde{M}t^{2}\|e\|_{2}^{2}. \end{align*} $

$M$足够大, 使得$C\|e\|^{4}-\lambda M\|e\|_{4}^{4}<0$, 则当$|t|\rightarrow \infty$时, 有$\Phi(te)\rightarrow -\infty.$

引理3.2  若(V), (K), (B), (g1)-(g4)成立, 则$\Phi$满足(PS)$_{c}$条件.

  设序列$\{u_{n}\}\subset E\backslash{0}$满足$\Phi(u_{n})\rightarrow c>0, \Phi'(u_{n})\rightarrow 0.$$n$充分大时, 有

$ \begin{align*} &c+1+\|u_{n}\|\geq\Phi(u_{n})-\frac{1}{4}\langle\Phi'(u_{n}), u_{n}\rangle\nonumber\\ =&\frac{1}{2}\|u_{n}\|^{2}+\frac{1}{4}\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u_{n}}u_{n}^{2}dx-\frac{1}{p+1}\displaystyle\int_{\mathbb{R}^{3}}b(x)|u_{n}|^{p+1}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}G(x, u)dx\nonumber\\ &-\frac{1}{4}(\|u_{n}\|^{2}+\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u_{n}}u_{n}^{2}dx-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u_{n}|^{p-1}u_{n}^{2}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}g(x, u_{n})u_{n}dx)\nonumber\\ =&\frac{1}{4}\|u_{n}\|^{2}+(\frac{1}{4}-\frac{1}{p+1})\|b\|^{1-p}_{2}\|u_{n}\|^{p+1}_{2}+\lambda(\displaystyle\int_{\mathbb{R}^{3}}\frac{1}{4}g(x, u_{n})u_{n}-G(x, u_{n}))dx\nonumber\\ \geq&\frac{1}{4}\|u_{n}\|^{2}+(\frac{1}{4}-\frac{1}{p+1})\|b\|^{1-p}_{2}\|u_{n}\|^{p+1}_{2}. \end{align*} $

再由$p\in(0, 1)$可知, $\{u_{n}\}$$E$中有界.

$E$中不妨设$u_{n}$弱收敛到$u$, 则在$L^{s}(\mathbb{R}^{3})~ (s\in(2, 6))$中, $u_{n}$强收敛到$u$, 且有

$ \begin{align*} \|u_{n}-u\|^{2}=&\langle I'(u_{n})-I'(u), u_{n}-u\rangle+\displaystyle\int_{\mathbb{R}^{3}}K(x)(\phi_{u}u-\phi_{u_{n}}u_{n})(u_{n}-u)dx\\ &+ \displaystyle\int_{\mathbb{R}^{3}}b(x)(|u_{n}|^{p-1}u_{n}-|u|^{p-1}u)dx+\lambda\displaystyle\int_{\mathbb{R}^{3}}(g(x, u_{n})-g(x, u))(u_{n}-u)dx. \end{align*} $

显然$ \langle I'(u_{n})-I'(u), u_{n}-u\rangle\rightarrow 0. $由Hölder不等式, 有

$ \begin{eqnarray*} &\left|\displaystyle\int_{\mathbb{R}^{3}}K(x)(\phi_{u}u-\phi_{u_{n}}u_{n})(u_{n}-u)dx\right|\leq(|K\phi_{u_{n}}|_{6}|u_{n}|_{2}+|K\phi_{u}|_{6}|u|_{2})|u_{n}-u|_{3}, \\ &\left|\displaystyle\int_{\mathbb{R}^{3}}b(x)(|u_{n}|^{p-1}u_{n}-|u|^{p-1}u)(u_{n}-u)dx\right|\leq|b|_{\frac{2}{1-p}}(|u_{n}|_{2}^{p}+|u|_{2}^{p})|u_{n}-u|_{2}, \\ &\left|\displaystyle\int_{\mathbb{R}^{3}}(g(x, u_{n})-g(x, u))(u_{n}-u)dx\right|\leq(|g(x, u_{n})|_{q+1}^{q}+|g(x, u)|_{q+1}^{q})|u_{n}-u|_{q+1}. \end{eqnarray*} $

由Sobolev不等式及$E$可以连续嵌入到$L^{s}(\mathbb{R}^{3})~ (s\in(2, 6))$中可知

$ \begin{eqnarray*} &\displaystyle\int_{\mathbb{R}^{3}}K(x)(\phi_{u}u-\phi_{u_{n}}u_{n})(u_{n}-u)dx\rightarrow 0, \\ &\displaystyle\int_{\mathbb{R}^{3}}b(x)(|u_{n}|^{p-1}u_{n}-|u|^{p-1}u)(u_{n}-u)dx\rightarrow 0, \\ &\displaystyle\int_{\mathbb{R}^{3}}(g(x, u_{n})-g(x, u))(u_{n}-u)dx\rightarrow 0. \end{eqnarray*} $

因此$\|u_{n}-u\|^{2}\rightarrow 0.$

引理 3.3  假设(V), (K), (B), (g1)--(g{4})成立, 则对任意的$u\in E\backslash\{0\}, $存在$t_{u}>0$, 使得$t_{u}u\in\mathcal{N}$.

  对任意给定的$u\in E\backslash\{0\}$, 由引理$3.1$中的(ⅱ)可知, 存在$R_{e}>0, $使得当$\|te\|\geq R_{e}$时, 有$\Phi(te)<0$.同理, 由引理$3.1$中的(ⅰ)可知, 对于$t>0$足够小时, 有$\Phi(tu)>0$.因此, $0<\max\Phi(tu)<\infty, $对于$t_{u}\in\mathbb{R}^{+}$, 有$\Phi(t_{u}u)=\max\Phi_{t\in \mathbb{R}^{+}}$.故$u_{0}=t_{u}u$$\Phi\mid_{\mathbb{R}^{+}u}$的一个临界点, 因此$\langle\Phi'(u_{0}), u_{0}\rangle=0$, 故$t_{u}u\in\mathcal{N}$.

引理 3.4  存在$\alpha_{0}>0$, 使得对所有的$u\in\mathcal{N}$, 有$\|u\|\geq \alpha_{0}.$

  因为$u\in\mathcal{N}$, 故$\langle\Phi'(u), u\rangle=0.$$\varepsilon>0$足够小, 则有

$ \begin{align*} 0&=\|u\|^{2}+\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u}u^{2}dx-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p-1}u^{2}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}g(x, u)udx\\ &\geq\|u\|^{2}-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u|^{p-1}u^{2}dx-\lambda\varepsilon|u|^{2}-\lambda C(\varepsilon)|u|^{q} \geq\frac{1}{2}\|u\|^{2}-C_{1}\|u\|^{q}. \end{align*} $

因此$\|u\|\geq\alpha_{0}>0, u\in\mathcal{N}$, 其中$\alpha_{0}=(\frac{1}{2C_{1}})^{\frac{1}{q-2}}$.

引理 3.5 [16]  设$r>0$, 如果$\{u_{n}\}$$H^{1}(\mathbb{R}^{3})$中有界, 且

$ \lim \limits_{n \rightarrow\infty}\sup \limits_{y\in\mathbb{R}^{3}}\displaystyle\int_{B_{r}(y)}|u_{n}|^{2}dx=0, $

则对于任意的$s\in(2, 6), $$L^{s}(\mathbb{R}^{3})$中有$u_{n}\rightarrow 0.$

定理1.1的证明  设$\{u_{n}\}\subset\mathcal{N}$$\Phi$的一个极小化序列, 且满足(PS)$_{c}$条件, 则$\|u_{n}\|$有界.令$c_{*}=\inf \limits_{\mathcal{N}}\Phi, \delta:=\lim \limits_{n \rightarrow\infty}\sup \limits_{y\in\mathbb{R}^{3}}\displaystyle\int_{B_{r}(y)}|u_{n}|^{2}dx$, 则$\delta>0.$如若不然, 则$\delta=0.$由引理$3.5$可知, 对任意的$s\in(2, 6), $$L^{s}(\mathbb{R}^{3})$中有$u_{n}\rightarrow 0.$从而当$n\rightarrow\infty$时, 有$\displaystyle\int_{\mathbb{R}^{3}}g(x, u_{n})u_{n}=o(\|u_{n}\|).$因此有

$ \begin{align*} o(\|u_{n}\|)=&\langle\Phi'(u_{n}), u_{n}\rangle\\ =&\|u\|^{2}+\displaystyle\int_{\mathbb{R}^{3}}K(x)\phi_{u_{n}}u_{n}^{2}dx-\displaystyle\int_{\mathbb{R}^{3}}b(x)|u_{n}|^{p-1}u_{n}^{2}dx-\lambda\displaystyle\int_{\mathbb{R}^{3}}g(x, u_{n})u_{n}dx\\ \geq&\|u_{n}\|^{2}-o(\|u_{n}\|). \end{align*} $

$\|u_{n}\|\rightarrow 0$, 这与引理$3.4$相矛盾, 故$\delta>0.$因此存在$r, \delta>0, \{y_{n}\}\subset\mathbb{Z}^{3}$, 使得

$ \lim \limits_{n \rightarrow\infty}\sup \limits_{y\in\mathbb{R}^{3}}\displaystyle\int_{B_{r}(y)}|u_{n}|^{2}dx\geq\delta>0. $

由(V), (g1)可知, 存在$u\in\mathcal{N}, $使得$u_{n}\rightharpoonup u\neq0$, 则$\Phi'(u)=0$.由于$u\in\mathcal{N}, $所以$\Phi(u)\geq c_{*}$.事实上, 由(g{4}), Fatou's引理(见文献[17]), $\|\cdot\|$的弱下半连续和$\{u_{n}\}$有界, 可得

$ \begin{align*} c_{*}+o(1)&=\Phi(u_{n})-\frac{1}{4}\langle\Phi'(u_{n}), u_{n}\rangle\\ &=\frac{1}{4}\|u_{n}\|^{2}+(\frac{1}{4}-\frac{1}{p+1})\|b\|^{1-p}_{2}\|u_{n}\|^{p+1}_{2}+\lambda(\displaystyle\int_{\mathbb{R}^{3}}\frac{1}{4}g(x, u_{n})u_{n}-G(x, u_{n}))dx+o(1)\\ &\geq\frac{1}{4}\|u\|^{2}+(\frac{1}{4}-\frac{1}{p+1})\|b\|^{1-p}_{2}\|u\|^{p+1}_{2}+\lambda(\displaystyle\int_{\mathbb{R}^{3}}\frac{1}{4}g(x, u)u-G(x, u))dx+o(1)\\ &=\Phi(u)-\frac{1}{4}\langle\Phi'(u), u\rangle+o(1)=\Phi(u)+o(1). \end{align*} $

所以$\Phi(u)\leq c_{*}$, 因此$\Phi(u)=c_{*}=\inf \limits_{\mathcal{N}}\Phi>0$, 证毕.

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