考虑如下方程

$\begin{eqnarray}\label{nbgftfrr} -\Delta u+V( x) u+\Big(u^2\star\frac{1}{|x|}\Big) u= |u|^{p-2}u- \mu|u|^{q-2}u ,\quad u\in H^1({\Bbb R}^3), \end{eqnarray}$

(1.1)

其中 $2＜p＜q＜6$,$\mu>0$ 是一个参数,

$$u^2\star\frac{1}{|x|}=\int_{{\Bbb R}^3}\frac{u^2(y)}{|x-y|}{\rm d}y,$$

$H^1({\Bbb R}^3)$ 为带有范数 $\| u\| =(\int_{{\Bbb R}^3}(|\nabla u|^2+u^2){\rm d}x)^{1/2}$的标准的Sobolev空间.

函数 $V\in L^\infty({\Bbb R}^3)$关于每个变量 $x_j$,$j = 1,2,3$ 都是以$1$为周期的周期函数. 在此条件下,算子

$\begin{eqnarray}\label{ncxbuc77dy} L=-\Delta+V: L^{2}({\Bbb R}^3)\rightarrow L^{2}({\Bbb R}^3) \end{eqnarray}$

(1.2)

的谱$\sigma(L)$ 是下方有界的连续谱,且可表示为无限多个互不相交的闭区间的并集 (参见文献[^[1],定理 XIII.100]).故${\Bbb R}\setminus \sigma(L)$ 由无穷多个互不相交的开区间构成,这些开区间被称为谱隙. 假设$V$满足

$(\bf{v})$~ $V\in L^\infty({\Bbb R}^3)$关于每个变量 $x_j$,$j = 1,2,3$ 都是以$1$为周期的周期函数且存在 $0＜\alpha,\beta＜+\infty$ 使得 $(-\alpha,\beta)$ 是 $-\Delta+V$的一个谱隙.

本文的主要结果是

定理1.1 设 $2＜p＜q＜6$满足 $2p>2+q$且$V$满足条件 $\bf ( v)$,则存在 $\mu_0>0$使对任意的 $\mu>\mu_0,$ 方程(1.1) 存在非零解.

方程 (1.1)来自原子物理学中的 Thomas-Fermi-Dirac-von Weizsäcker (TFDW)理论. 在文献^[2]中,Lieb 对不同类型的函数$V$研究了方程

$$-A\Delta\psi+(V+\alpha)\psi+(\psi^2\star |x|^{-1})\psi+\gamma\psi^{2p-1}-C_e\psi^{\frac{5}{3}} =-\mu\psi,\quad\mbox{在$ {\Bbb R}^3$中} $$

解的存在性和对称性,其中,$A,\alpha,\gamma,C_e$和 $\mu$ 为常数且 $p>4/3.$ 在文献^[3]中,Le Bris 研究了最小值问题

$\begin{equation}\label{mm0suuyfhsss} \inf_{\psi\in H^1({\Bbb R}^3),\ \int_{{\Bbb R}^3}\psi^2{\rm d}x=\lambda}E(\psi) , \end{equation}$

(1.3)

其中

$\begin{eqnarray*} E(\psi) & = & \int_{{\Bbb R}^3}|\nabla\psi|^2{\rm d}x+\int_{{\Bbb R}^3}V(x)\psi^2{\rm d}x+c_1\int_{{\Bbb R}^3}\psi^\frac{10}{3}{\rm d}x -c_2\int_{{\Bbb R}^3}\psi^\frac{8}{3}{\rm d}x\\ & & +\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{\psi^2(x)\psi^2(y)}{|x-y|}{\rm d}x, \end{eqnarray*}$

其中 $c_1$,$c_2$ 为正常数,$V$为 Coulomb位势函数. 他证明了 存在 $\lambda_c>0$使当 $0＜\lambda＜\lambda_c$时,该极小值被达到. 此外, 若 $ c_2$充分小,则达到函数是唯一的. 容易验证存在$\mu\in {\Bbb R}$使得达到函数满足

$$-\Delta\psi+V(x)\psi+(\psi^2\star|x|^{-1})\psi+c_1\psi^{\frac{7}{3}}-c_2\psi^{\frac{5}{3}}=\mu\psi, \quad\mbox{在${\Bbb R}^3$中.} $$

然而,在最近的文献 ^[4]中,作者证明了当$V=0$且 $\lambda$充分大时,极小值 (1.3)式不能被达到. 最后, 方程 (1.1)还与如下的 Schr\"odinger-Poisson-Slater 方程有密切联系(参见文献 ^[5-18]等)

$$-\Delta u+V( x) u+\Big(u^2\star\frac{1}{|x|}\Big) u= |u|^{p-2}u,\quad \mbox{在${\Bbb R}^3$中.} $$

方程 (1.1)的变分泛函为

$\begin{eqnarray}\label{ncxbttdref} J_{\mu}(u) & = & \frac{1}{2}\int_{{\Bbb R}^3}(|\nabla u|^2+V(x)u^2){\rm d}x +\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y\\ & & -\frac{1}{p}\int_{{\Bbb R}^3}|u|^p{\rm d}x+\frac{\mu}{q}\int_{{\Bbb R}^3}|u|^q{\rm d}x,\ u\in H^1({\Bbb R}^3). \end{eqnarray}$

(1.4)

$J_\mu$的临界点为方程 (1.1)的解. 本文运用 文献^[19]中的一个新的无穷维环绕定理得到了方程(1.1)的一个非零解.

本文结构安排如下: 第二节给出方程(1.1)的变分结构; 第三节证明$-J_\mu$具有整体环绕几何结构; 第四节证明$-J_\mu$的$(\overline{C})_c$序列是有界的,第五节给出定理1.1的证明.

记号 $B_r(a)$表示以 $r$为半径以$a$为中心的开球. Banach空间 $X$的对偶空间记为 $X'$, $X$中的强收敛和弱收敛分别记为 $\rightarrow$ 和 $\rightharpoonup$. 设 $\varphi\in C^1(X;\mathbb{R}),$ $\varphi$ 在$u$ 的Fr\'echet导数记为 $\varphi'(u)$. $\varphi$的 Gateaux导数记为 $\langle \varphi'(u),v\rangle,$ $\forall u,v\in X.$ $L^q({\Bbb R}^3)$ 和 $L^q_{loc}({\Bbb R}^3)$ 为标准的 $L^q$ 空间和局部可积的 $L^q$ 空间,其中$1\leq q\leq\infty$.

令

$$D^{1,2}({\Bbb R}^3)=\bigg\{u\in L^6({\Bbb R}^3)\ \bigg |\ \int_{{\Bbb R}^3}|\nabla u|^2{\rm d}x＜\infty\bigg\}.$$

它是一个Hilbert空间,其内积为

$$(u,v)_{D^{1,2}({\Bbb R}^3)}=\int_{{\Bbb R}^3}\nabla u\nabla v{\rm d}x.$$

对$u\in H^1({\Bbb R}^3)$,由 Lax-Milgram定理,方程

$\begin{eqnarray}\label{ncb88fyf11}-\Delta\phi=4\pi u^2,\quad \mbox{在${\Bbb R}^3$中} \end{eqnarray}$

(2.1)

有唯一解 $\phi_u\in D^{1,2}({\Bbb R}^3)$ (见文献[^[20],性质 2.2]). 由文献[^[21],定理2.2.1], $\phi_u$有如下表达式

$\begin{eqnarray}\label{nvjguutytr} \phi_u(x)=\Big(u^2\star\frac{1}{|y|}\Big)(x)=\int_{{\Bbb R}^3} \frac{u^2(y)}{|x-y|}{\rm d}y,\quad x\in {\Bbb R}^3. \end{eqnarray}$

(2.2)

函数 $\phi_u$有如下性质(见文献^[6]):

引理2.1 (i) 存在正常数 $C$,使对任意的 $u\in H^1({\Bbb R}^3)$,有

$\begin{eqnarray}\label{mvnuufyfh} \| \phi_u\| _{D^{1,2}({\Bbb R}^3)}\leq C\| u\| ^2_{H^1({\Bbb R}^3)} \end{eqnarray}$

(2.3)

且

$\begin{eqnarray}\label{bchhdgdt8u} \int_{{\Bbb R}^3}|\nabla\phi_u|^2{\rm d}x & = & 4\pi\int_{{\Bbb R}^3}\phi_u u^2{\rm d}x =4\pi\int_{{\Bbb R}^{3}}\int_{{\Bbb R}^{3}}\frac{u^{2}(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y\nonumber\\ & \leq & C\| u\| ^4_{L^{12/5}({\Bbb R}^3)}\leq C\| u\| ^4_{H^1({\Bbb R}^3)}. \end{eqnarray}$

(2.4)

(ii) 对任意的 $u\in H^1({\Bbb R}^3)$,在${\Bbb R}^3$上$\phi_u\geq 0$.

由该引理可知泛函 $J_\mu$在$H^1({\Bbb R}^3)$上的定义是良好的且

$\begin{eqnarray}\label{bc88dytr22} \langle J'_{\mu}(u),v\rangle & = & \int_{{\Bbb R}^3}(\nabla u\nabla v+V(x)uv){\rm d}x+\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u(x)v(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y \nonumber\\ & & -\int_{{\Bbb R}^3}|u|^{p-2}uv{\rm d}x+\mu\int_{{\Bbb R}^3}|u|^{q-2}uv{\rm d}x,\ \forall u,v\in H^1({\Bbb R}^3). \end{eqnarray}$

(2.5)

故 $J_\mu$为$H^1({\Bbb R}^3)$上的$C^1$泛函且

引理2.2 如下命题等价

(i) $u\in H^1({\Bbb R}^3)$ 为方程(1.1)的解;

(ii) $u$为$J_{\mu}$的临界点.

令

$$X=H^{1}({\Bbb R}^3).$$

由文献^[22]的第六章6.4节可知存在正交分解$X=Y\oplus Z$,其 中 $Y$,$Z$为$X$的闭子空间且在$Z^{3}$的平移作用下不变,使得

$\begin{equation}\label{hnbxxc5r8871z} \forall u\in Y,\ \int_{{\Bbb R}^3}(|\nabla u|^{2}+Vu^{2}){\rm d}x=(u,u)=\| u\| ^{2},\end{equation}$

(2.6)

$\begin{equation}\label{un77715490kjh76} \forall u\in Z,\ \int_{{\Bbb R}^3}(|\nabla u|^{2}+Vu^{2}){\rm d}x=-(u,u)=-\| u\| ^{2}. \end{equation}$

(2.7)

令$Q:X\rightarrow Z$,$P:X\rightarrow Y$为正交投影算子. 由 (2.6)和 (2.7)式得

$\begin{eqnarray}\label{vd66dtrdpp} \int_{{\Bbb R}^{3}}(|\nabla u|^{2}+Vu^{2}){\rm d}x=\| Pu\| ^2-\| Qu\| ^2,\ \forall u\in X. \end{eqnarray}$

(2.8)

此外,还有

$\begin{eqnarray}\label{nfhyfyrttr11} u=Pu+Qu,\quad \| u\| ^2=\| Pu\| ^2+\| Qu\| ^2,\quad \forall u\in X. \end{eqnarray}$

(2.9)

由(2.8)和 (1.4)式得

$\begin{eqnarray}\label{ncxbtsdsesqf} -J_{\mu}(u) & = & \frac{1}{2}\| Qu\| ^2-\frac{1}{2}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y \nonumber\\ & & +\frac{1}{p}\int_{{\Bbb R}^3}|u|^p{\rm d}x-\frac{\mu}{q}\int_{{\Bbb R}^3}|u|^q{\rm d}x,\ u\in X. \end{eqnarray}$

(2.10)

此外,由 (2.5)式得

$\begin{eqnarray}\label{hdgtdrddd} \langle-J'_{\mu}(u),v\rangle & = & (Qu,v)-(Pu,v) -\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u(x)v(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y\nonumber\\ & & +\int_{{\Bbb R}^3}|u|^{p-2}uv{\rm d}x-\mu\int_{{\Bbb R}^3}|u|^{q-2}uv{\rm d}x,\ \forall u,v\in X. \end{eqnarray}$

(2.11)

设$\{e_k\}$ 为$Y$的一个标准正交基.令

$\begin{eqnarray}\label{nvcooiugyu} \| |u\| |=\max\bigg\{\| Qu\| ,\sum^{\infty}_{k=1}\frac{1}{2^{k+1}}|(Pu,e_k)|\bigg\}. \end{eqnarray}$

(3.1)

设 $R > r > 0$,$u_0 \in Z$满足$\| u_0\| = 1$,令

$$ N = \{u \in Z \ |\ \| u\| = r\},\ M = \{u+tu_0\ |\ u\in Y ,\ t\geq 0,\ \| u+tu_0\| \leq R\}, $$

$$ \partial M=\{u\in Y\ |\ \| u\| \leq R\}\cup\{u+tu_0\ |\ u\in Y,\ t>0,\ \| u+tu_0\| =R \}. $$

定义3.1 设 $\phi\in C^1(X,\mathbb{R})$. 若对任意的 $\varphi\in X$以及对满足$u_n\rightharpoonup u$的任意的$u\in X$和 $\{u_n\}\subset X$,有 $\langle \phi'(u_n),\varphi\rangle\rightarrow \langle \phi'(u),\varphi\rangle$成立,则称 $\phi'$是弱序列连续的.

引理3.1 泛函 $-J_{\mu}$满足如下条件

(a) $-J'_{\mu}$是弱序列连续的且对任意的 $\mu>0,$

$\begin{eqnarray}\label{qam99iuxyyxy} \sup_M(-J_{\mu})＜+\infty; \end{eqnarray}$

(3.2)

(b) 存在 $\delta>0,$ $R > r > 0$,$\mu'_0>0$ 以及 满足 $\| u_0\| =1$的$u_0 \in Z $ 使对任意的 $\mu>\mu'_0,$

$\begin{eqnarray}\label{qanx99s8s7yy} \inf_N (-J_{\mu})>\max\Big\{ \sup_{\partial M} (-J_{\mu}), \sup_{\| |u\| |\leq\delta}(-J_{\mu}(u))\Big\}. \end{eqnarray}$

(3.3)

证 (a) 设$u\in X$,$\{u_n\}\subset X$满足 $u_n\rightharpoonup u$,则

$\begin{eqnarray}\label{hd66dtrffq} (Qu_n,v)\rightarrow (Qu,v),\ (Pu_n,v)\rightarrow (Pu,v),\ n\rightarrow\infty,\forall v\in X. \end{eqnarray}$

(3.4)

由 $u_n\rightharpoonup u$得 $u_n\rightarrow u$ 在 $L^s_{loc}({\Bbb R}^3)$中,$\forall 1\leq s＜6$,故对任意的$v\in C^\infty_0({\Bbb R}^3),$ 有

$\begin{eqnarray}\label{nfdhhhhh8y} \int_{{\Bbb R}^3}|u_n|^{p-2}u_nv{\rm d}x\rightarrow\int_{{\Bbb R}^3}|u|^{p-2}uv{\rm d}x,\quad \int_{{\Bbb R}^3}|u_n|^{q-2}u_nv{\rm d}x\rightarrow\int_{{\Bbb R}^3}|u|^{q-2}uv{\rm d}x. \end{eqnarray}$

(3.5)

由引理 2.1可知$\{\phi_{u_n}\}$在 $D^{1,2}({\Bbb R}^3)$中有界. 故不妨设 $\phi_{u_n}\rightharpoonup\phi_{u}$. 从而对任意的$1\leq s＜6$,在 空间$L^s_{loc}({\Bbb R}^3)$中 $\phi_{u_n}\rightarrow\phi_{u}$.由此可知,对任意的 $v\in C^\infty_0({\Bbb R}^3),$有

$\begin{eqnarray}\label{mvn88fyrttr} \int_{{\Bbb R}^3}\phi_{u_n}u_nv{\rm d}x\rightarrow\int_{{\Bbb R}^3}\phi_{u}uv{\rm d}x. \end{eqnarray}$

(3.6)

由 (2.11),(3.4),(3.5)式 以及(3.6)式可得

$$\langle-J'_{\mu}(u_n),v\rangle\rightarrow \langle-J'_{\mu}(u), v\rangle,\ \forall v\in C^\infty_0({\Bbb R}^3).$$

故 $-J'_{\mu}$是弱序列连续的. 此外,容易看出$\sup_{M}(-J_{\mu})＜+\infty$.

(b)~ 由 (2.4)和 (2.10)式得: 对任意的 $u\in Z,$有

$\begin{eqnarray}\label{ncuudyfh} -J_{\mu}(u) & \geq & \frac{1}{2}\| u\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y -\frac{\mu}{q}\int_{{\Bbb R}^3}|u|^q{\rm d}x\nonumber\\ & \geq & \frac{1}{2}\| u\| ^2-\frac{C}{16\pi}\| u\| ^4-\frac{C_1\mu}{q}\| u\| ^q,\nonumber \end{eqnarray}$

其中 $C_1$为常数.故存在充分小的 $r>0$ 使得

$\begin{equation}\label{cb99dudy} \inf_{N}(-J_{\mu})>\frac{1}{4}r^2>0. \end{equation}$

(3.7)

由于 $q>p$,故存在 $\Lambda>0$使当 $\mu\geq 1$,$\quad |t|>\Lambda$时

$\begin{equation}\label{mvnhhdtr66rt11} \frac{1}{p}|t|^p-\frac{\mu}{q}|t|^q＜0 . \end{equation}$

(3.8)

设 $C'>0$满足

$\begin{equation}\label{mciiiicjpp09} \| u\| _{L^2({\Bbb R}^3)}\leq C'\| u\| ,\ \forall u\in X. \end{equation}$

(3.9)

由于$q>p>2$,故存在 $\mu'_0>1$ 使对任意的 $\mu>\mu'_0,$

$\begin{equation}\label{nv00dufhhhaa} \sup_{|t|\leq \Lambda}t^{-2}\Big(\frac{1}{p}|t|^p-\frac{\mu}{q}|t|^q\Big)＜\frac{1}{4C'^2}. \end{equation}$

(3.10)

由 (3.8) 和 (3.10)式得: 对任意的 $\mu>\mu'_0$和 $u\in X,$有

$\begin{eqnarray}\label{bcgttf88uy} -J_{\mu}(u) & = & \frac{1}{2}\| Qu\| ^2-\frac{1}{2}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y +\int_{{\Bbb R}^3} \Big(\frac{1}{p}|u|^p-\frac{\mu}{q}|u|^q\Big){\rm d}x \nonumber\\ & \leq & \frac{1}{2}\| Qu\| ^2-\frac{1}{2}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y \\ & & +\int_{\{x\ |\ |u(x)|\leq \Lambda\}} \Big(\frac{1}{p}|u|^p-\frac{\mu}{q}|u|^q\Big){\rm d}x\nonumber\\ & \leq & \frac{1}{2}\| Qu\| ^2-\frac{1}{2}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y +\frac{1}{4C'^2}\int_{{\Bbb R}^3}u^2{\rm d}x\nonumber\\ & \leq & \frac{1}{2}\| Qu\| ^2-\frac{1}{2}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y +\frac{1}{4}\| u\| ^2\nonumber\\ & = & \frac{3}{4}\| Qu\| ^2-\frac{1}{4}\| Pu\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y. \end{eqnarray}$

(3.11)

由文献[^[23],定理1.1]可知,存在 $C_*>0$ 使对任意的 $u\in X,$有

$\begin{eqnarray}\label{nv99iuyfhdd} \int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y\geq C_*\Big(\int_{{\Bbb R}^3}\frac{u^2}{|x|^{1/2}(1+|\ln|x\| )}{\rm d}x\Big)^2. \end{eqnarray}$

(3.12)

设 $u_0\in Z$满足$\| u_0\| =1.$ 令 $u=v+tu_0\in Y\oplus Ru_0.$ 由于存在连续投影

$$L^2({\Bbb R}^3,|x|^{-1/2}(1+|\ln|x\| )^{-1} )\rightarrow R u_0,$$

故存在常数 $C_{**}>0$使对任意的$u=v+tu_0\in Y\oplus Ru_0,$有

$\begin{eqnarray}\label{mnu877zkigugjhh} \Big(\int_{{\Bbb R}^3}\frac{u^2}{|x|^{1/2}(1+|\ln|x\| )}{\rm d}x\Big)^2\geq C_{**}t^4. \end{eqnarray}$

(3.13)

由 (3.11)--(3.13)式可知,存在常数 $\kappa>0$使对任意的 $u=tu_0+v\in Y\oplus Ru_0$,有

$\begin{eqnarray}\label{mnghgyuyu00}-J_{\mu}(u)\leq \frac{3}{4}t^2-\frac{1}{4}\| v\| ^2-\kappa t^4.\nonumber \end{eqnarray}$

故当$\| u\| \rightarrow\infty$且$ u\in Y\oplus Ru_0$时

$\begin{eqnarray}\label{vdgttdg112} -J_{\mu}(u)\rightarrow -\infty. \end{eqnarray}$

(3.14)

由 (3.11) 式可得当 $u\in Y$时

$\begin{eqnarray}\label{md090d9d8ww} -J_{\mu}(u)\leq-\frac{1}{4}\| u\| ^2-\frac{1}{4}\int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u^2(x)u^2(y)}{|x-y|}{\rm d}x{\rm d}y\leq 0.\nonumber \end{eqnarray}$

联合(3.14)式可知存在$R>r$使得

$\begin{eqnarray}\label{mv99fugfytt} \sup_{\partial M}(-J_{\mu})\leq 0＜\inf_N(-J_{\mu}). \end{eqnarray}$

(3.15)

由(3.11)式和$\| |\cdot\| |$ 的定义(见 (3.1)式)可知当 $\mu>\mu'_0$时

$\begin{eqnarray}\label{ncx99cifyuhh} -J_{\mu}(u) \leq \frac{3}{4}\| Qu\| ^2\leq\frac{3}{4}\| |u\| |^2. \end{eqnarray}$

(3.16)

取$\delta=r/\sqrt{6}$,由(3.16)和 (3.7)式得

$$\sup_{\| |u\| |\leq\delta}(-J_{\mu}(u))＜\inf_{N}(-J_{\mu}).$$

联合 (3.15)式我们得到(3.3)式. 证毕.

定义4.1 设 $\phi\in C^1(X,\mathbb{R})$. 若 $\{u_n\} \subset X$ 满足

$$ \sup_n\phi(u_n) \leq c,\ (1+\| u_n\| )\| \phi'(u_n)\| _{X'}\rightarrow 0,\ n\rightarrow\infty,$$

则称其为 $\phi$ 的一个$(\overline{C})_c$序列.

由$q>p>2$可知存在与$\mu$无关的常数 $\gamma>2$,当$\mu>>1$时,对

$\begin{equation}\label{ssxnv9911q} |t|>D_\mu:=\gamma\mu^{-1/(q-p)}, \end{equation}$

(4.1)

有

$\begin{equation}\label{nv99ifufu11q} \mu|t|^{q-2}- |t|^{p-2}>\| V_-\| _{L^\infty({\Bbb R}^3)} , \end{equation}$

(4.2)

其中 $V_-(x)=\max\{-V(x),0\}$.

引理4.1 设 $\mu\geq 1.$ 若 $\{u_n\}$ 为$-J_{\mu}$的一个 $(\overline{C})_c$序列, 则

$\begin{eqnarray}\label{nmciifu77t7} \lim_{n\rightarrow\infty}\int_{\{x\ |\ |u_n(x)|>D_\mu\}}|u_n|^6{\rm d}x=0. \end{eqnarray}$

(4.3)

证 令 $v_n=\max\{u_n(x)-D_\mu/2,0\}$. 易证 $v_n\in X$ 且 $\| v_n\| \leq \| u_n\| ,$ $\forall n\in\mathbb{N}.$ 由 $\{u_n\}$ 为$-J_{\mu}$的一个 $(\overline{C})_c$序列 得 $\langle J'_\mu(u_n),v_n\rangle=o(1)$. 在(2.5)式中取 $v=v_n$,则由 (4.2)式以及 $\phi_{u_n},$ $u_n$ 和 $v_n$为 $\{x\ |\ u_n(x)>D_\mu/2\}$上的非负函数的事实可得

$\begin{eqnarray}\label{mv99vhfyyf55r} o(1) & = & \langle J'_\mu(u_n), v_n\rangle \\ & = & \int_{{\Bbb R}^3}|\nabla v_n|^2{\rm d}x+\int_{{\Bbb R}^3}V(x) u_nv_n{\rm d}x+\int_{{\Bbb R}^3}\phi_{u_n}\cdot u_n\cdot v_n{\rm d}x\nonumber\\ & & +\int_{{\Bbb R}^3}(\mu|u_n|^{q-2}-|u_n|^{p-2})u_nv_n{\rm d}x\nonumber\\ & = & \int_{{\Bbb R}^3}|\nabla v_n|^2{\rm d}x+\int_{{\Bbb R}^3}V_+ u_nv_n{\rm d}x+\int_{\{x\ |\ u_n(x)>D_\mu/2\}}\phi_{u_n}\cdot u_n\cdot v_n{\rm d}x\nonumber\\ & & +\int_{\{x\ |\ u_n(x)>D_\mu/2\}}(\mu|u_n|^{q-2}-|u_n|^{p-2}-V_-(x))u_nv_n{\rm d}x\nonumber\\ & \geq & \int_{{\Bbb R}^3}|\nabla v_n|^2{\rm d}x \geq \widetilde{C}\Big(\int_{{\Bbb R}^3}| v_n|^6{\rm d}x\Big)^{\frac{1}{3}}\\ & = & \widetilde{C}\Big(\int_{\{x\ |\ u_n(x)>D_\mu/2\}}| v_n|^6{\rm d}x\Big)^{\frac{1}{3}}, \end{eqnarray}$

(4.4)

其中 $V_+=V+V_-\geq 0$ 为 ${\Bbb R}^3$上非负函数,$\widetilde{C}$ 为Sobolev常数.注意到在$\{x\ |\ u_n(x)>D_\mu\}$上 $v_n\geq u_n/2$. 则由 (4.4)式可得

$$\lim_{n\rightarrow\infty}\int_{\{x\ |\ u_n(x)>D_\mu\}}| u_n|^6{\rm d}x=0.$$

同理可得

$$\lim_{n\rightarrow\infty}\int_{\{x\ |\ -u_n(x)>D_\mu\}}| u_n|^6{\rm d}x=0.$$

证毕.

引理4.2 存在 $\mu''_0>0$使得当 $\mu>\mu''_0$时, 若$\{u_n\}$为$-J_{\mu}$的一个 $(\overline{C})_c$ 序列,则 $\{u_n\}$在$X$中有界.

证 由 $(1+\| u_n\| )\| -J'_{\mu}(u_n)\| _{X'}\rightarrow 0$得

$\begin{eqnarray}\label{bc88duyysaa} \langle -J'_{\mu}(u_n),Qu_n\rangle=o(1),\ \langle -J'_{\mu}(u_n),Pu_n\rangle=o(1). \end{eqnarray}$

(4.5)

在(2.11)式中取 $v=Qu_n$,$v=Pu_n$,由 (4.5)式得

$$ \| Qu_k\| ^2=\int_{{\Bbb R}^3}u_n\phi_{u_n}\cdot Qu_n{\rm d}x-\int_{{\Bbb R}^3}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot Qu_n{\rm d}x+o(1)\nonumber $$

及

$$ \| Pu_n\| ^2=-\int_{{\Bbb R}^3}u_n\phi_{u_n}\cdot Pu_n{\rm d}x+\int_{{\Bbb R}^3}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n) \cdot Pu_n{\rm d}x+o(1).\nonumber $$

再由 $\| u_n\| ^2=\| Pu_n\| ^2+\| Qu_n\| ^2$ (见 (2.9)式)得

$\begin{eqnarray}\label{hf55drzm} \| u_n\| ^2 & = & \int_{{\Bbb R}^3}u_n\cdot\phi_{u_n}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & -\int_{{\Bbb R}^3}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot(Qu_n-Pu_n){\rm d}x+o(1). \end{eqnarray}$

(4.6)

令

$\begin{eqnarray} \chi_{n,1}(x)=\left\{\begin{array} [c]{ll} 1 , & \mbox{若 } \ |u_n(x)|\leq D_\mu,\\ 0, & \mbox{若 }\ |u_n(x)|> D_\mu, \end{array} \right. \end{eqnarray}$

(4.7)

其中 $D_\mu$来自 (4.1)式. 设 $\chi_{n,2}=1-\chi_{n,1}.$ 则 $u^2_n=(u_n\chi_{n,1})^2+(u_n\chi_{n,2})^2$ 且

$\begin{eqnarray}\label{nx99xusyttdd} \phi_{u_n}=\phi_{u_n\chi_{n,1}}+\phi_{u_n\chi_{n,2}}. \end{eqnarray}$

(4.8)

由于 $\phi_{u_n\chi_{n,1}}$ 为${\Bbb R}^3$上方程

$$-\Delta\phi=4\pi (u_n\chi_{n,1})^2$$

的解,由 $(u_n\chi_{n,1})^2\leq D_\mu^2$以及标准的椭圆方程正则性理论可知存在与 $n$ 和 $y\in {\Bbb R}^3$无关的正常 数 $C_2$使对任意的 $y\in {\Bbb R}^3$,有

$\begin{eqnarray}\label{nnbvyyfh765} \| \phi_{u_n\chi_{n,1}}\| _{L^\infty(B_1(y))}\leq C_2\Big(\int_{B_2(y)}|u_n\chi_{n,1}|^4{\rm d}x\Big)^{\frac{1}{2}}\leq C_2D_\mu^2\Big(\int_{B_2(y)}{\rm d}x\Big)^{\frac{1}{2}}. \end{eqnarray}$

(4.9)

记$C_3= (\int_{B_2(0)}{\rm d}x )^{\frac{1}{2}}.$ 由 (4.9)式得

$$ \| \phi_{u_n\chi_{n,1}}\| _{L^\infty({\Bbb R}^3)}\leq C_2C_3 D_\mu^2. $$

再由 $\phi_{u_n\chi_{n,1}}$为 ${\Bbb R}^3$上非负函数 (见引理 2.1 (ii))得

$\begin{eqnarray}\label{1qnc99iewtdgg} 0\leq\phi_{u_n\chi_{n,1}}\leq C_2C_3 D_\mu^2,\quad \mbox{在${\Bbb R}^3$中.} \end{eqnarray}$

(4.10)

由 (4.6),(4.8) 式以及(4.10)式得

$\begin{eqnarray} \| u_n\| ^2 & = & \int_{{\Bbb R}^3}u_n\cdot\phi_{u_n\chi_{n,1}}\cdot(Qu_n-Pu_n){\rm d}x+\int_{{\Bbb R}^3}u_n\cdot\phi_{u_n\chi_{n,2}}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & -\int_{\{x\ |\ |u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & -\int_{\{x\ |\ |u_n(x)|> D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot(Qu_n-Pu_n){\rm d}x+o(1)\nonumber\\ & \leq & \| \phi_{u_n\chi_{n,1}}u_n\| _{L^2({\Bbb R}^3)}\| Qu_n-Pu_n\| _{L^2({\Bbb R}^3)} +\int_{{\Bbb R}^3}u_n\cdot\phi_{u_n\chi_{n,2}}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & +\Big(\int_{\{x\ |\ |u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| Qu_n-Pu_n\| _{L^2({\Bbb R}^3)}\nonumber\\ & & +\Big(\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{p}{\rm d}x\Big)^{\frac{p-1}{p}}\| Qu_n-Pu_n\| _{L^p({\Bbb R}^3)}\nonumber\\ & & +\mu\Big(\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{q}{\rm d}x\Big)^{\frac{q-1}{q}}\| Qu_n-Pu_n\| _{L^q({\Bbb R}^3)}.\nonumber \end{eqnarray}$

再由 $\| Qu_n-Pu_n\| _{L^2({\Bbb R}^3)}\leq 2C'\| u_n\| $, $\| Qu_n-Pu_n\| _{L^p({\Bbb R}^3)}\leq C''\| u_n\| $以及 $\| Qu_n-Pu_n\| _{L^q({\Bbb R}^3)}\leq C''\| u_n\| $得

$\begin{eqnarray*} \| u_n\| ^2 & \leq & 2C'\| u_n\phi_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}\| u_n\| +\int_{{\Bbb R}^3}u_n\cdot\phi_{u_n\chi_{n,2}}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & +2C'\Big(\int_{\{x\ |\ |u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| u_n\| \nonumber\\ & & + C''\Big(\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{p}{\rm d}x\Big)^{\frac{p-1}{p}}\| u_n\|\\ & & +\mu C''\Big(\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{q}{\rm d}x\Big)^{\frac{q-1}{q}}\| u_n\| .\nonumber \end{eqnarray*}$

由引理 4.1可知当$\mu\geq1$时有 $\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{p}{\rm d}x=o(1)$,$\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{q}{\rm d}x=o(1)$. 由此以及 (4.10)式,从上述不等式可得当$\mu\geq1$时,有

$\begin{eqnarray}\label{bhfttdfrrd} \| u_n\| ^2 & \leq & 2C'D_\mu(C_2C_3)^{\frac{1}{2}}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}\| u_n\| +\int_{{\Bbb R}^3}u_n\cdot\phi_{u_n\chi_{n,2}}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & & +2C'\Big(\int_{\{x\ |\ |u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| u_n\|\\ & & +o(1)\cdot\| u_n\| . \end{eqnarray}$

(4.11)

由 Hardy-Littlewood-Sobolev不等式(见文献[^[24],(1.1)式])得

$\begin{eqnarray}\label{bchgyftd55wes} & & \int_{{\Bbb R}^3}u_n\cdot \phi_{u_n\chi_{n,2}}\cdot (Qu_n-Pu_n){\rm d}x\nonumber\\ & = & \int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{(u_n\chi_{n,2}(y))^2\cdot u_n(x)\cdot(Qu_n-Pu_n)(x)}{|x-y|}{\rm d}x{\rm d}y\nonumber\\ & \leq & C_{HLS}\| (u_n\chi_{n,2})^2\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\| u_n\cdot(Qu_n-Pu_n)\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\nonumber\\ & \leq & C_{HLS}\| (u_n\chi_{n,2})^2\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\| u_n\| _{L^{\frac{12}{5}}({\Bbb R}^3)}\| Qu_n-Pu_n\| _{L^{\frac{12}{5}}({\Bbb R}^3)} \nonumber\\ & \leq & C'''\Big(\int_{\{x\ |\ |u_n(x)|> D_\mu\}}|u_n|^{\frac{12}{5}}{\rm d}x\Big)^{\frac{5}{6}}\| u_n\| ^2, \end{eqnarray}$

(4.12)

其中 $C_{HLS}$ 和 $C'''$为正常数. 由引理 4.1可知,当$\mu\geq 1$时,有

$\begin{eqnarray}\label{bc909144243}\int_{\{x\ |\ |u_n(x)|> D_\mu \}}|u_n|^{\frac{12}{5}}{\rm d}x=o(1). \end{eqnarray}$

(4.13)

由(4.13),(4.12)式以及 (4.11)式 可得,当$\mu\geq 1$时,有

$\begin{eqnarray}\label{bhfttdfrrdh661} \| u_n\| ^2 & \leq & 2C'D_\mu(C_2C_3)^{\frac{1}{2}}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}\| u_n\| +o(1)\cdot\| u_n\| ^2\nonumber\\ & & +2C'\Big(\int_{\{x\ |\ |u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| u_n\|\\ & & +o(1)\cdot\| u_n\| . \end{eqnarray}$

(4.14)

设$0＜\epsilon＜D_\mu$. 则对$\mu\geq 1,$有

$\begin{eqnarray}\label{hfyyrterrq} & & \Big(\int_{\{x\ |\ |u_n(x)|\leq \epsilon \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\nonumber\\ & \leq & \mu\Big(\int_{\{x\ |\ |u_n(x)|\leq \epsilon \}}(|u_n|^{p-1}+|u_n|^{q-1})^2{\rm d}x\Big)^{\frac{1}{2}}\nonumber\\ & \leq & 2\mu\Big(\int_{\{x\ |\ |u_n(x)|\leq \epsilon \}}(|u_n|^{2p-2}+|u_n|^{2q-2}){\rm d}x\Big)^{\frac{1}{2}}\nonumber\\ & \leq & 4\mu(\epsilon^{p-2}+\epsilon^{q-2}) \Big(\int_{\{x\ |\ |u_n(x)|\leq \epsilon \}}|u_n|^{2}{\rm d}x\Big)^{\frac{1}{2}}\\ & \leq & 4\mu C'(\epsilon^{p-2}+\epsilon^{q-2})\| u_n\| . \end{eqnarray}$

(4.15)

由 (4.15)和(4.14)式可得, 当$\mu\geq 1$时,有

$\begin{eqnarray}\label{bc88fuyydhd11} \| u_n\| ^2 & \leq & 2C'D_\mu(C_2C_3)^{\frac{1}{2}}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}\| u_n\| +o(1)\cdot\| u_n\| ^2\\ & & +4 \mu C'^2(\epsilon^{p-2}+\epsilon^{q-2})\| u_n\| ^2\nonumber\\ & & +2C'\Big(\int_{\{x\ |\ \epsilon＜|u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| u_n\| \\ & & +o(1)\cdot\| u_n\| . \end{eqnarray}$

(4.16)

由于$2p>2+q$,故存在与$\mu$无关的正常数$1＜\beta＜\gamma$使得

$$\epsilon_\mu=\beta\mu^{-1/(q-p)}$$

满足

$\begin{eqnarray}\label{1xxsbc88fywsdyydws} 4\mu C'^2(\epsilon^{p-2}_\mu+\epsilon^{q-2}_\mu)＜1/2,\ \epsilon_\mu＜D_\mu. \end{eqnarray}$

(4.17)

则由 (4.16)式得

$\begin{eqnarray}\label{bc88fywsdyydws} \| u_n\| ^2 & \leq & 4C'D_\mu(C_2C_3)^{\frac{1}{2}}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}\| u_n\| +o(1)\cdot\| u_n\| ^2+o(1)\cdot\| u_n\| \nonumber\\ & & +4C'\Big(\int_{\{x\ |\ \epsilon_\mu＜|u_n(x)|\leq D_\mu \}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\Big)^{\frac{1}{2}}\| u_n\| . \end{eqnarray}$

(4.18)

由$\sup_n(-J_{\mu}(u_n))\leq c$ 以及 $(1+\| u_n\| )\| -J'_{\mu}(u_n)\| _{X'}\rightarrow 0$得

$\begin{eqnarray}\label{nnchfdgdttdte} o(1)+c & \geq & -J_{\mu}(u_{n})+\frac{1}{2}\langle J'_{\mu}(u_{n}), u_n\rangle\nonumber\\ & = & \frac{1}{4}\int_{{\Bbb R}^3}u^2_n\phi_{u_n}{\rm d}x -\int_{{\Bbb R}^3} \Big(\Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n|^p-\mu \Big(\frac{1}{2}-\frac{1}{q}\Big)|u_n|^q\Big){\rm d}x. \end{eqnarray}$

(4.19)

故有

$\begin{eqnarray}\label{nc88dmmmaaz} & & \frac{1}{4}\int_{{\Bbb R}^3}u^2_n\phi_{u_n}{\rm d}x +\int_{\{x\ |\ |u_n(x)|>\mu^{-1/(q-p)}\}} \Big(\mu\Big(\frac{1}{2}-\frac{1}{q}\Big)|u_n(x)|^q- \Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n(x)|^p\Big){\rm d}x\nonumber\\ & \leq & c+\int_{\{x\ |\ |u_n(x)|\leq\mu^{-1/(q-p)}\}} \Big(\Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n|^p-\mu\Big(\frac{1}{2}-\frac{1}{q} \Big)|u_n|^q\Big){\rm d}x+o(1)\nonumber\\ & \leq & c+ C_4\mu^{-\frac{q-2}{q-p}}\int_{{\Bbb R}^3}|u_n|^2{\rm d}x+o(1), \end{eqnarray}$

(4.20)

其中 $C_4$ 为正常数. 由于 $\varepsilon_\mu>\mu^{-1/(q-p)}$,故

$\begin{eqnarray} & & \int_{\{x\ |\ D_\mu\geq|u_n(x)|>\epsilon_\mu\}} \Big(\mu\Big(\frac{1}{2}-\frac{1}{q}\Big)|u_n|^q- \Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n|^p\Big){\rm d}x\nonumber\\ & \leq & \int_{\{x\ |\ |u_n(x)|>\mu^{-1/(q-p)}\}}\Big(\mu\Big(\frac{1}{2}-\frac{1}{q}\Big)|u_n|^q- \Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n|^p\Big){\rm d}x.\nonumber \end{eqnarray}$

则由 (4.20) 式和 $\int_{{\Bbb R}^3}u^2_n\phi_{u_n}{\rm d}x\geq\int_{{\Bbb R}^3}u^2_n\phi_{u_n\chi_{n,1}}{\rm d}x=\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| ^2_{L^2({\Bbb R}^3)}$可得

$\begin{eqnarray}\label{qanc88dmmmaaz} & & \frac{1}{4}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}^2 +\int_{\{x\ |\ D_\mu\geq|u_n(x)|>\epsilon_\mu\}} \Big(\mu\Big(\frac{1}{2}-\frac{1}{q}\Big)|u_n|^q- \Big(\frac{1}{2}-\frac{1}{p}\Big)|u_n|^p\Big){\rm d}x\nonumber\\ & \leq & c+ C_4\mu^{-\frac{q-2}{q-p}}\int_{{\Bbb R}^3}|u_n|^2{\rm d}x+o(1)\nonumber\\ & \leq & c+ C'^2C_4\mu^{-\frac{q-2}{q-p}}\| u_n\| ^2+o(1). \end{eqnarray}$

(4.21)

容易验证存在正常数$C_\mu\geq C\mu^{(p-2)/(q-p)}$ 使得 对任意的 $\epsilon_\mu\leq |t|\leq D_\mu$有

$$C_\mu(|t|^{p-2}t-\mu|t|^{q-2}t)^2\leq \Big(\frac{1}{2}-\frac{1}{q}\Big)\mu|t|^q-\Big(\frac{1}{2}-\frac{1}{p}\Big)|t|^p,$$

其中$C>0$为与$\mu$无关的正常数. 则由 (4.21)式得

$\begin{eqnarray}\label{nchuu4dydttw} & & \frac{1}{4}\| u_n\phi^{\frac{1}{2}}_{u_n\chi_{n,1}}\| _{L^2({\Bbb R}^3)}^2 +C_\mu\int_{\{x\ |\ D_\mu\geq|u_n(x)|>\epsilon_\mu\}}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)^2{\rm d}x\nonumber\\ & \leq & c+ C'^2C_4\mu^{-\frac{q-2}{q-p}}\| u_n\| ^2+o(1). \end{eqnarray}$

(4.22)

由 (4.22)和(4.18)式可知存在 $\mu''_0>0$, 使当$\mu>\mu''_0$时,$\{\| u_n\| \}$ 有界. 证毕.

定理1.1的证明 由引理 3.1,引理 4.2以及文献[^[19],定理 1.3]可知存在$\mu_0>0$ 使得 当$\mu>\mu_0$时,存在 $-J_\mu$的有界 $(\overline{C})_c$序列 $\{u_n\}$ 满足 $\inf_n\| |u_n\| |>0$. 则$\{u_n\}$满足下列两个条件之一:

(a)~ $\lim\limits_{n\rightarrow\infty}\sup\limits_{y\in {\Bbb R}^3}\int_{B_1(y)}|u_n|^2{\rm d}x=0$;

(b)~ 存在 $\varrho>0$以及 $a_n\in Z^3$ 使得 $\int_{B_1(a_n)}|u_n|^2{\rm d}x\geq\varrho.$ 若(a)成立,则由Lions引理 (见文献[^[22],定理 1.21]) 可知对任意的$2＜s＜6$,在$L^s({\Bbb R}^3)$中$u_n\rightarrow 0$. 故

$\begin{equation}\label{nc00sowwww} \int_{{\Bbb R}^3}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot Qu_n{\rm d}x\rightarrow 0, \end{equation}$

(5.1)

$\begin{equation}\label{nc00sowwww1} \int_{{\Bbb R}^3}(|u_n|^{p-2}u_n-\mu|u_n|^{q-2}u_n)\cdot Pu_n{\rm d}x\rightarrow 0. \end{equation}$

(5.2)

仿(4.12)式的证明可得

$\begin{eqnarray}\label{n00dfdhfgfff} & & \int_{{\Bbb R}^3}u_n\cdot\phi_{u_n}\cdot(Qu_n-Pu_n){\rm d}x\nonumber\\ & = & \int_{{\Bbb R}^3}\int_{{\Bbb R}^3}\frac{u_n^2(y)\cdot u_n(x)\cdot(Qu_n-Pu_n)(x)}{|x-y|}{\rm d}x{\rm d}y\nonumber\\ & \leq & C_{HLS}\| u_n^2\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\| u_n\cdot(Qu_n-Pu_n)\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\nonumber\\ & \leq & C_{HLS}\| u_n^2\| _{L^{\frac{6}{5}}({\Bbb R}^3)}\| u_n\| _{L^{\frac{12}{5}}({\Bbb R}^3)}\| Qu_n-Pu_n\| _{L^{\frac{12}{5}}({\Bbb R}^3)} \nonumber\\ & = & C_{HLS}\| u_n\| ^3_{L^{\frac{12}{5}}({\Bbb R}^3)}\| Qu_n-Pu_n\| _{L^{\frac{12}{5}}({\Bbb R}^3)}\rightarrow 0,\ n\rightarrow\infty. \end{eqnarray}$

(5.3)

由 (5.1)--(5.3)式以及 (4.6)式得 $\| u_n\| \rightarrow 0.$这与 $\inf_{n}\| |u_n\| |>0$矛盾. 故 (a)不成立. 从而只能 (b)成立.令 $w_n=u_n(\cdot+a_n)$则 $w_n\rightharpoonup u_0\neq 0$. 由 $(1+\| w_n\| )\| -J'_{\mu}(w_n)\| _{X'}=(1+\| u_n\| )\| -J'_{\mu}(u_n)\| _{X'}\rightarrow 0$ 以及 $-J'_{\mu}$的弱序列连续性得 $-J'_{\mu}(u_0)=0.$ 故 $u_0$ 为 方程 (1.1)的一个非零解. 证毕.