本文主要研究如下分数阶Schrödinger-Kirchhoff方程多重解的存在性

(1.1) $ \begin{equation} M\bigg({{\int\!\!\!\int}}_{{{\Bbb R}} ^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg)(-\Delta)_p^{s} u+V(x)|u|^{p-2}u = f(x, u)+\lambda h(x)|u|^{r-2}u, \; x\in {{{\Bbb R}} }^N, \end{equation} $

其中$ \lambda\in {{\Bbb R}} $, $ 0<s<1<r<p<2 $, $ ps<N $. $ (-\Delta)_p^{s} $表示分数阶$ p $ -拉普拉斯算子, 即对任给定$ \varphi\in C_0^{\infty}({{{\Bbb R}} }^N) $, 有

$ (-\Delta)_p^{s}\varphi(x) = 2\lim\limits_{\epsilon\rightarrow 0^+}\int_{{{{\Bbb R}} }^N\backslash B_{\epsilon}(x)}\frac{|\varphi(x)-\varphi(y)|^{p-2}(\varphi(x)-\varphi(y))}{|x-y|^{N+ps}}{\rm d}y, \ \ \ \ x\in {{{\Bbb R}} }^N, $

其中$ B_{\epsilon}(x) = \{y\in {{{\Bbb R}} }^N: |x-y|<\epsilon\} $.更多关于分数阶$ p $ -拉普拉斯算子的定义和性质参见文献[1, 2].近年来, 许多学者开始研究物理背景很强的Kirchhoff方程

$ \begin{eqnarray*} \label{eq:a83} u_{tt}-\bigg(a+b\int_{\Omega}|\nabla u|^2{\rm d}x\bigg)\Delta u = g(x, u), \ \ \ x\in \Omega, \end{eqnarray*} $

其中$ \Omega $是$ {{{\Bbb R}} }^N $中的光滑区域, $ a>0 $, $ b\geq 0 $.该问题作为著名D'Alembert波方程

$ \rho \frac{{{\partial ^2}u}}{{\partial {t^2}}} - (\frac{{{p_0}}}{\lambda } + \frac{E}{{2L}}\int_0^L | \frac{{\partial u}}{{\partial x}}{|^2}{\rm{d}}x)\frac{{{\partial ^2}u}}{{\partial {x^2}}} = g(x, u) $

的推广在1883年被Kirchhoff提出. Kirchhoff模型考虑了弦横向振动产生的弦长变化, 其中$ L $代表弦的长度, $ g $表示截面面积, $ E $表示材料的杨氏模, $ \rho $表示质量密度, $ p_0 $表示初始张力. Lions在1978年建立了Kirchhoff方程的抽象研究框架.自此各类Kirchhoff型方程逐渐成为国内外研究的热点对象.更多关于Kirchhoff型方程的研究参见文献[3-6].另一方面, 相对于微分算子$ -\Delta $来说, 分数阶拉普拉斯算子是一类非局部拟微分算子.由于非局部微分方程较经典的局部微分算子能够更好、更详实地描述物理实验现象, 因此分数阶$ p $ -拉普拉斯方程解的相关问题研究已成为非线性分析领域的热门研究方向之一.

关于分数阶Kirchhoff型方程, Fiscella和Valdinoci在文献[7]中首先在$ {{{\Bbb R}} }^{N} $的有界区域上提出了一个基于Kirchhoff的变分模型, 该模型考虑了弦的分数长度的非局部测量引起的张力的变化.在文献[8]中, 作者利用Ekeland变分原理和山路定理研究了如下带有分数阶$ p $ -拉普拉斯算子的Schrödinger-Kirchhoff型方程多重解的存在性

$ M\bigg({{\int\!\!\!\int}}_{ {{\Bbb R}} ^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg)(-\Delta)_p^{s} u+V(x)|u|^{p-2}u = f(x, u)+g(x), \quad x\in {{{\Bbb R}} }^N, $

其中扰动项$ g:{{{\Bbb R}} }^N \rightarrow {{\Bbb R}} $. Zhang和Luo等人在文献[9]中研究了如下分数阶的Schrödinger-Kirchhoff方程

$ \bigg(1+\alpha{{\int\!\!\!\int}}_{{{\Bbb R}} ^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{3+2s}}{\rm d}x{\rm d}y\bigg)(-\Delta)^{s} u+u = \beta f(u)+u^{2_{s}^{*}-1}, \quad x\in {{{\Bbb R}} }^3, $

其中$ s\in (0, 1) $, $ 2_{s}^{*} = \frac{6}{3-2s} $.在文献[9]中, 作者首先利用Pohozǎev不等式, 在参数$ \alpha $, $ \beta $和$ f(u) $满足一定条件下, 得到上述问题基态解的存在性.其次, 当$ s\in (0, \frac{3}{4}] $时, 作者给出了解不存在的相关结论.文献[10]研究了如下分数阶的Krichhoff-Schrödinger-Poisson系统多重解的存在性

$ \begin{eqnarray*} \left\{\begin{array}{ll} { } M\bigg({{\int\!\!\!\int}}_{ {{\Bbb R}} ^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg)(-\Delta)^{s} u+V(x)u+\phi(x)u = \lambda f(x, u), \ & x\in {{{\Bbb R}} }^3, \\ (-\Delta)^{t}\phi(x) = u^2, \ & x\in {{{\Bbb R}} }^3, \end{array}\right. \end{eqnarray*} $

其中$ s, t\in (0, 1) $, $ 2t+4s>3 $.在文献[10]中, 作者利用喷泉定理和对称山路定理给出了上述系统无穷多解的存在性.在文献[11]中, 作者同样利用喷泉定理得到了一类有界区域上分数阶Krichhoff型方程无穷解的存在性.

受上述文献以及文献[12-17]的启发, 本文主要利用变分法和临界点理论研究一类带有分数阶$ p $ -拉普拉斯算子的Schrödinger-Kirchhoff方程多重解的存在性.类似于文献[11, 16]中关于Kirchhoff函数$ M $和势函数$ V $的假设, 下面给出本文所需要假设:

(M1) $ M\in C({{{\Bbb R}} }^+, {{{\Bbb R}} }^+) $, 并且存在常数$ m_0>0 $, 使得$ \inf_{t\in {{{\Bbb R}} }^+} M(t)\geq m_0>0 $;

(M2) 存在$ \theta\in [1, \frac{N}{N-sp}) $, 使得对一切$ t\in {{{\Bbb R}} }^+ $, 成立$ \theta \overline{M}(t) = \theta\int_0^tM(\tau){\rm d}\tau\geq M(t)t $;

(V) $ V\in C({{{\Bbb R}} }^N, {{\Bbb R}} ) $, $ {V_0}: = \mathop {\inf }\limits_{x \in {\mathbb{R}^N}} {\mkern 1mu} V\left( x \right) > 0 $且$ \mathop {\lim }\limits_{\left| x \right| \to + \infty } V\left( x \right) = + \infty $;

(f1) $ f\in C({{{\Bbb R}} }^N\times {{\Bbb R}} , {{\Bbb R}} ) $, 且存在常数$ C_1>0 $使得$ |f(x, u)|\leq C_1(|u|^{p-1}+|u|^{q-1}) $, 其中$ \theta p<q<p_s^* = \frac{Np}{N-sp} $;

(f2) 存在$ \mu>\theta p $和$ l>0 $, 使得当$ |u|\geq l $时, $ 0<\mu F(x, u): = \mu\int_0^u f(x, t){\rm d}t\leq f(x, u)u $, 并且$ \mathop {\inf }\limits_{x \in {\mathbb{R}^N}, |u| = l} {\mkern 1mu} F(x, u) > 0 $;

(f3) $ \forall x\in {{{\Bbb R}} }^N $, $ u\in {{\Bbb R}} $, 成立$ f(x, -u) = -f(x, u) $;

(h) $ h: {{{\Bbb R}} }^N \rightarrow {{{\Bbb R}} }^+ $, 并且$ h\in L^{\frac{p}{p-r}}({{{\Bbb R}} }^N) $.

文中所得主要结果如下:

定理1.1   假设条件(M1), (M2), (V), (f1)–(f3), (h)成立, 则存在$ \lambda_0>0 $, 当$ \lambda<\lambda_0 $时, 方程(1.1)有无穷多解, 且对应的能量值趋于无穷大.

定理1.2   假设条件(M1), (M2), (V), (f1)–(f3), (h)成立, 则对任意的$ \lambda>0 $, 方程(1.1)有一列能量值趋于0的负能量解.

设$ 1\leq \nu<\infty $, $ L^{\nu}({{{\Bbb R}} }^N) $表示通常的$ \nu $次Lebesgue可积函数空间, 对应范数为

$ \|u\|_{\nu} = \bigg(\int_{{{{\Bbb R}} }^N}|u|^{\nu} {\rm d}x\bigg)^{\frac{1}{\nu}}. $

给定实数$ s\in (0, 1) $, 分数阶Sobolev空间$ W^{s, p}({{{\Bbb R}} }^N) $定义为

$ W^{s, p}({{{\Bbb R}} }^N): = \big\{u\in L^p({{{\Bbb R}} }^N): [u]_{s, p}<\infty\big \}, \ \ \hbox{其中}\ [u]_{s, p} = \bigg({{\int\!\!\!\int}}_{ {{{\Bbb R}} }^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\bigg)^{\frac{1}{p}}, $

赋予范数

(2.1) $ \begin{equation} \|u\|_{W^{s, p}} = \big(\|u\|_{p}^p+[u]_{s, p}^p\big)^{\frac{1}{p}}. \end{equation} $

对于方程(1.1), 作用空间$ E $定义为

$ E: = \bigg\{u\in W^{s, p}({{{\Bbb R}} }^N): \int_{{{{\Bbb R}} }^N}V(x)|u(x)|^p{\rm d}x<\infty\bigg \}, $

赋予范数

$ \|u\| = \big([u]_{s, p}^p+\|u\|_{p, V}^p\big)^{\frac{1}{p}}, \ \ \hbox{其中}\ \|u\|_{p, V}^p = \int_{{{{\Bbb R}} }^N}V(x)|u(x)|^p{\rm d}x. $

$ L^r({{{\Bbb R}} }^N, h)(1<r<2) $表示加权的Lebesgue空间, 即

$ L^r({{{\Bbb R}} }^N, h) = \bigg\{u: {{{\Bbb R}} }^N\rightarrow {{\Bbb R}} \ \hbox{是可测函数, 且}\ \int_{{{{\Bbb R}} }^N}h(x)|u(x)|^r{\rm d}x<+\infty\bigg\}, $

赋予范数

$ \|u\|_{L^{r}({{{\Bbb R}} }^N, h)} = \bigg(\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^r{\rm d}x\bigg)^{\frac{1}{r}}. $

显然方程(1.1)对应的泛函$ I:E\rightarrow {{\Bbb R}} $定义为

$ I(u) = \frac{1}{p}\big(\overline{M}([u]_{s, p}^p)+\|u\|_{p.V}^p\big)-\int_{{{{\Bbb R}} }^N}F(x, u){\rm d}x-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x. $

由文献[8, 17]易证, $ I\in C^1(E, {{\Bbb R}} ) $, 并且对任意$ u, v\in E $, 有

$ \begin{eqnarray*} \label{eq:a28} \langle I'(u), v\rangle & = &M([u]_{s, p}^p){{\int\!\!\!\int}}_{ {{{\Bbb R}} }^{2N}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\\ &&+ \int_{{{{\Bbb R}} }^N}V(x)|u|^{p-2}uv{\rm d}x-\int_{{{{\Bbb R}} }^N}f(x, u)v{\rm d}x-\lambda\int_{{{{\Bbb R}} }^N}h(x)|u|^{r-2}uv{\rm d}x. \end{eqnarray*} $

为了方便, 文中用$ C $表示任意正常数.

定义2.1   如果对任意的$ \varphi\in E $, 有

$ \begin{align} & M([u]_{s, p}^{p})\iint_{{{\mathbb{R}}^{2N}}}{\frac{|u(x)-u(y){{|}^{p-2}}(u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y{{|}^{N+ps}}}}\text{d}x\text{d}y \\ & +\int_{{{\mathbb{R}}^{N}}}{V}(x)|u{{|}^{p-2}}u\varphi \text{d}x-\int_{{{\mathbb{R}}^{N}}}{f}(x, u)\varphi \text{d}x-\lambda \int_{{{\mathbb{R}}^{N}}}{h}(x)|u{{|}^{r-2}}u\varphi \text{d}x=0 \\ \end{align} $

成立, 则称$ u\in E $是方程(1.1)的弱解.

引理2.1^[18]   假设条件$ (V) $成立, $ s\in(0, 1) $, $ \nu\in [p, p_{s}^{*}] $, 则$ W^{s, p}({{{\Bbb R}} }^N) $连续嵌入到$ L^{\nu}({{{\Bbb R}} }^N) $, 即存在Sobolev嵌入常数$ M_{\nu}>0 $, 使得

$ \|u\|_{\nu}\leq M_{\nu} \|u\|_{W}, \ \ \ u\in W^{s, p}({{{\Bbb R}} }^N). $

根据文献[8], 当$ \nu\in [p, p_{s}^{*}) $时, 空间$ E $紧嵌入到$ L^{\nu}({{{\Bbb R}} }^N) $.

因为$ E $是自反的Bananch空间, 故存在$ E $的一列可数基$ \{e_i\}_{i = 1}^{\infty} $, 令$ X_i = Re_i $. $ Y_k = \bigoplus\limits_{i = 1}^{k}X_i, \ Z_k = \overline{\bigoplus\limits_{i = k+1}^{\infty}X_i}, \ k\in \mathbb{N} $, 显然$ E = Y_k\oplus Z_k $.

定义2.2^[19]   泛函$ I\in C^1(E, {{\Bbb R}} ) $满足能量水平为$ c\in{{\Bbb R}} $的Palais-Smale条件(简记为$ (PS)_c $条件)是指任何满足

(2.2) $ \begin{equation} I(u_n)\rightarrow c, \ \ \ I'(u_n)\rightarrow 0, \ \ \ \ n\rightarrow \infty \end{equation} $

的序列$ \{u_n\}\subset E $都存在一个收敛子序列.任何满足(2.2)式的序列叫做$ (PS)_c $序列.

定义2.3^[19]   泛函$ I\in C^1(E, {{\Bbb R}} ) $满足能量水平为$ c\in {{\Bbb R}} $的$ (PS)^{*}_c $条件(关于$ Y_n $), 是指任何满足

$ u_{n_j}\in Y_{n_j}, \ \ \ \ I(u_{n_j})\rightarrow c, \ \ \ \ I|'_{Y_{n_j}}\rightarrow 0, \ \ \ \ n_j\rightarrow \infty $

的序列$ \{u_{n_j}\}\subset E $都存在一个收敛子序列.

定理2.1 (对称山路定理^[20])   设$ E $是一个无穷维Banach空间, $ E = Y\oplus Z $, 其中$ \hbox{dim} Y<\infty $. $ I\in C^1(E, {{\Bbb R}} ) $是偶泛函满足$ (PS)_c $条件, $ I(0) = 0 $, 并且

(A1) 存在常数$ \rho $, $ \alpha>0 $, 使得$ I_{\partial B_{\rho}\cap Z} = \inf\limits_{u\in Z, \|u\| = \rho}I(u) \geq \alpha $;

(A2) 对$ E $的任意有限维子空间$ E'\subset E $, 存在常数$ R = R(E') $使得$ \max\limits_{u\in E', \|u\|\geq R}I(u)<0 $.

那么, $ I $有一列无界的临界点序列.

定理2.2 (对偶喷泉定理^[21])   设偶泛函$ I\in C^1(E, {{\Bbb R}} ) $, $ k_0\in \mathbb{N} $.如果对任意的$ k>k_0 $, 存在$ r_k>\gamma_k>0 $使得

(B1) $ a_k = \inf\{I(u): u\in Z_k, \|u\| = r_k\}\geq 0 $;

(B2) $ b_k = \max\{I(u): u\in Y_k, \|u\| = \gamma_k\}< 0 $;

(B3) $ d_k = \inf\{I(u): u\in Z_k, \|u\|\leq r_k\}\rightarrow 0 $, $ k\rightarrow \infty $;

(B4) 对任意的$ c\in [d_{k_0}, 0] $, $ I $满足$ (PS)^{*}_c $条件.

那么, $ I $有一列收敛到$ 0 $的负临界点序列.

引理3.1   假设(M1), (M2), (V), (f1)–(f3), (h)成立.则对任意的$ c\in {{\Bbb R}} $, 泛函$ I $满足$ (PS)_c $条件.

证   首先证明$ (PS)_c $序列$ \{u_n\} $有界.事实上, 由条件(h)可推得

$ \int_{{{{\Bbb R}} }^N}h(x)|u_n(x)|^r{\rm d}x\leq \|h\|_{\frac{p}{p-r}}\|u\|_p^r\leq M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r. $

再根据假设(f2), 存在常数$ l>0 $, 使得$ f(x, u)u\geq \mu F(x, u), |u|\geq l, $并且, 对任意给定的$ 0<C_0<(\frac{1}{4p}-\frac{1}{4\mu})V_0 $, 存在常数$ \delta>0 $, 使得对任意的$ |u|\leq \delta $, 有

$ \Big|\frac{1}{\mu}f(x, u)u-F(x, u)\Big|\leq C_0 |u|^p. $

结合条件(f1), 当$ \delta\leq|u|\leq l $时, 有

$ \Big|\frac{1}{\mu}f(x, u)u-F(x, u)\Big|\leq C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)|u|^p. $

因此, 对任意的$ |u|\leq l $, 有

(3.1) $ \begin{equation} \Big|\frac{1}{\mu}f(x, u)u-F(x, u)\Big|\leq C_0|u|^p+C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)|u|^p. \end{equation} $

又因为$ \lim\limits_{|x|\rightarrow +\infty}V(x) = +\infty $, 即存在$ R>l>0 $使

$ \Big(\frac{1}{4p}-\frac{1}{4\mu}\Big)V(x)>C \Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big), \ \ |x|\geq R. $

结合(3.1)式, 得

(3.2) $ \begin{eqnarray} &&\Big(\frac{1}{p}-\frac{1}{\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x+\int_{|u_n(x)|\leq l} \Big[\frac{1}{\mu}f(x, u_n)u_n-F(x, u_n)\Big]{\rm d}x \\ & \geq&\Big(\frac{1}{p}-\frac{1}{\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x-\int_{|u_n(x)|\leq l} \Big[C_0|u_n|^p+C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)|u_n|^p\Big]{\rm d}x \\ & = & \Big(\frac{1}{2p}-\frac{1}{2\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x-\int_{|u_n(x)|\leq l} C_0|u_n|^p{\rm d}x+ \Big(\frac{1}{2p}-\frac{1}{2\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&-\int_{|u_n(x)|\leq l}C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)|u_n|^p{\rm d}x \\ & \geq& \int_{|u_n(x)|\leq l}\Big[\Big(\frac{1}{4p}-\frac{1}{4\mu}\Big)V_0-C_0\Big]|u_n|^p{\rm d}x+ \Big(\frac{1}{2p}-\frac{1}{2\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&-C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)l^p \cdot {\rm meas}\{x\in {{{\Bbb R}} }^N \big| |x|\leq R\} \\ & \geq &\Big(\frac{1}{2p}-\frac{1}{2\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x-C \Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)l^p \cdot {\rm meas}\{x\in {{{\Bbb R}} }^N \big| |x|\leq R\}. \end{eqnarray} $

由$ \{u_n\} $为$ (PS)_c $序列知, 当$ n $充分大时, 有

$ \begin{eqnarray*} c+\|u_n\|&\geq& I(u_n)-\frac{1}{\mu}\langle I'(u_n), u_n\rangle\\ &\geq& \Big (\frac{1}{p\theta}-\frac{1}{\mu}\Big)m_0[u_n]_{s, p}^p+ \Big(\frac{1}{p}-\frac{1}{\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&+\int_{{{{\Bbb R}} }^N}\Big[\frac{1}{\mu}f(x, u_n)u_n-F(x, u_n)\Big]{\rm d}x -\Big(\frac{1}{r}-\frac{1}{\mu}\Big)\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r \\ &\geq& \Big (\frac{1}{p\theta}-\frac{1}{\mu}\Big)m_0[u_n]_{s, p}^p+ \Big(\frac{1}{p}-\frac{1}{\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&+\int_{|u_n(x)|\leq l}\Big[\frac{1}{\mu}f(x, u_n)u_n-F(x, u_n)\Big]{\rm d}x- \Big(\frac{1}{r}-\frac{1}{\mu}\Big)\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r, \end{eqnarray*} $

故

$ \begin{eqnarray*} c+\|u_n\| &\geq&\Big (\frac{1}{p\theta}-\frac{1}{\mu}\Big)m_0[u_n]_{s, p}^p+ \Big(\frac{1}{p}-\frac{1}{\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&+\int_{|u_n(x)|\leq r}\Big[\frac{1}{\mu}f(x, u_n)u_n-F(x, u_n)\Big]{\rm d}x -\Big(\frac{1}{r}-\frac{1}{\mu}\Big)\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r\\ &\geq&\Big (\frac{1}{p\theta}-\frac{1}{\mu}\Big)m_0[u_n]_{s, p}^p+ \Big (\frac{1}{2p}-\frac{1}{2\mu}\Big)\int_{{{{\Bbb R}} }^N}V(x)|u_n|^p{\rm d}x\\ &&-C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)l^p \cdot {\rm meas}\{x\in {{{\Bbb R}} }^N \big| |x|\leq R\} -\Big(\frac{1}{r}-\frac{1}{\mu}\Big)\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r.\\ &\geq& \min\bigg\{\Big(\frac{1}{p\theta}-\frac{1}{\mu}\Big)m_0, \frac{1}{2p}-\frac{1}{2\mu}\bigg\} \|u_n\|^p\\ &&-C\Big(\frac{1}{\delta^{p-1}}+l^{q-p}\Big)l^p \cdot {\rm meas}\{x\in {{{\Bbb R}} }^N \big| |x|\leq R\} -\Big(\frac{1}{r}-\frac{1}{\mu}\Big)\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u_n\|^r. \end{eqnarray*} $

即$ \{u_n\} $是$ E $中有界序列.根据引理2.1可知, 存在$ u\in E $以及$ \{u_n\} $的子列(仍记为$ \{u_n\} $)满足:

在$ E $中, $ u_n\rightharpoonup u $;

在$ L^{\nu}({{{\Bbb R}} }^N) $, $ (p\leq \nu< p_s^*) $中$ u_n \rightarrow u $;

对几乎处处的$ x\in {{{\Bbb R}} }^N $, $ u_n(x)\rightarrow u(x) $.

为了证明$ \{u_n\} $在$ E $中强收敛到$ u $, 设$ \varphi\in E $, 定义线性泛函$ B_\varphi:E\rightarrow {{\Bbb R}} $如下

$ B_{\varphi}(v) = {{\int\!\!\!\int}}_{ {{{\Bbb R}} }^{2N}}\frac{|\varphi(x)-\varphi(y)|^{p-2}(\varphi(x)-\varphi(y))} {|x-y|^{N+ps}}(v(x)-v(y)){\rm d}x{\rm d}y. $

根据Hölder不等式, 可知

$ |B_{\varphi}(v)|\leq \|\varphi\|^{p-1}\|v\|, \ \ \ \ \forall v\in E, $

即$ B_{\varphi} $是连续的.结合$ \{M([u_n]_{s, p}^p)-M([u]_{s, p}^p)\}_n $的有界性可知

$ \lim\limits_{n\rightarrow \infty}(M([u_n]_{s, p}^p)-M([u]_{s, p}^p))B_u(u_n-u) = 0. $

最后, 结合Hölder不等式, 条件(f1)和(h)易证

$ \lim\limits_{n\rightarrow \infty}\int_{{{{\Bbb R}} }^N}(f(x, u_n)-f(x, u))(u_n-u){\rm d}x = 0. $

$ \lim\limits_{n \rightarrow \infty}\int_{{{{\Bbb R}} }^N}(h(x)|u_n|^{r-2}u_n-h(x)|u|^{r-2}u)(u_n-u){\rm d}x = 0. $

综上所述, 当$ n\rightarrow \infty $时, 有

$ \begin{eqnarray*} o(1)& = &\langle I'(u_n)-I'(u), u_n-u\rangle \\ & = & M([u_n]_{s, p}^p)B_{u_n}(u_n-u)-M([u]_{s, p}^p)B_u(u_n-u)\\ &&+\int_{{{{\Bbb R}} }^N}V(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u){\rm d}x+ \int_{{{{\Bbb R}} }^N}(f(x, u_n)-f(x, u))(u_n-u){\rm d}x\\ &&+\lambda\int_{{{{\Bbb R}} }^N}h(x)(|u_n|^{r-2}u_n-|u|^{r-2}u)(u_n-u){\rm d}x\\ & = & M([u_n]_{s, p}^p)[B_{u_n}(u_n-u)-B_u(u_n-u)]\\ &&+\int_{{{{\Bbb R}} }^N}V(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u){\rm d}x+o(1), \end{eqnarray*} $

即

$ \begin{eqnarray*} &&\lim\limits_{n\rightarrow \infty}\big( M([u_n]_{s, p}^p)[B_{u_n}(u_n-u)-B_u(u_n-u)]\\ &&+\int_{{{{\Bbb R}} }^N}V(x)(|u_n|^{p-2}u_n-|u|^{p-2}u)(u_n-u){\rm d}x\big) = 0. \end{eqnarray*} $

类似文献[8]中引理6的证明, $ \{u_n\} $在$ E $中强收敛到$ u $, 即$ I $满足$ (PS)_c $条件.

引理3.2   假设(M1), (M2), (V), (f1)–(f3), (h)成立.则对任意的$ c\in {{\Bbb R}} $, $ I $满足$ (PS)_c^{*} $条件.

证   根据定义2.3, 只需证明:对任意的$ c\in {{\Bbb R}} $, $ \{u_{n_j}\}\subset E $, 并且$ u_{n_j}\in Y_{n_j}, \ I(u_{n_j})\rightarrow c, \ I|'_{Y_{n_j}}\rightarrow 0 $, $ n_j\rightarrow \infty $, $ \{u_{n_j}\} $必存在收敛子序列.证明方法和引理3.1完全类似.

引理3.3   设$ p\leq \nu<p_s^{*} $.对任意的$ k\in \mathbb{N} $, 令

$ \beta_\nu(k): = \sup \big\{\|u\|_{\nu}: u\in Z_k, \|u\| = 1\big\}, $

则当$ k\rightarrow \infty $时, $ \beta_\nu(k)\rightarrow 0 $.

证   类似于文献[22]可证.首先根据定义可知$ 0<\beta_\nu(k+1)\leq \beta_\nu(k) $, 所以存在$ \beta>0 $使得$ \beta_\nu(k)\rightarrow \beta $, $ (k\rightarrow \infty) $.其次, 对任意的$ k\geq 0 $, 存在$ u_k\in Z_k $, $ \|u_k\| = 1 $, $ \|u_k\|_\nu>\frac{\beta_\nu(k)}{2} $.根据$ Z_k $的定义可得$ u_k\rightharpoonup 0 $.结合引理2.1可知$ u_k\rightarrow 0 $, 即$ \beta = 0 $.

定理1.1的证明   根据引理3.1可知对任意的$ c\in {{\Bbb R}} $, $ I $满足$ (PS)_c $条件.下面证明$ I $满足对称山路定理的条件(A1), (A2).根据假设条件(f1), 有

$ \begin{eqnarray*} |F(x, u)|\leq \frac{C_1}{p}|u|^p+\frac{C_1}{q}|u|^q, \ \ \forall(x, u)\in {{{\Bbb R}} }^N\times {{\Bbb R}} . \end{eqnarray*} $

再根据假设(M1), (h)和引理2.1, 知

$ \begin{eqnarray*} I(u)& = &\frac{1}{p}\big(\overline{M}([u]_{s, p}^p)+\|u\|_{p.V}^p\big)-\int_{{{{\Bbb R}} }^N}F(x, u){\rm d}x-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ &\geq & \frac{\min\{1, m_0\}}{p}\|u\|^p- \frac{C_1}{p}\int_{{{{\Bbb R}} }^N}|u|^p{\rm d}x- \frac{C_1}{q}\int_{{{{\Bbb R}} }^N}|u|^q{\rm d}x-\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u\|^{r}\\ &\geq & \frac{\min\{1, m_0\}}{p}\|u\|^p-C_1M_p^p\|u\|^p- C_1M_q^q\|u\|^q-\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u\|^{r}\\ & = & \|u\|^p\bigg[\Big(\frac{\min\{1, m_0\}}{p}-C_1M_p^p\Big)- C_1M_q^q\|u\|^{q-p}-\lambda M_p^r\|h\|_{\frac{p}{p-r}}\|u\|^{r-p}\bigg]. \end{eqnarray*} $

设

$ \eta(t) = \bigg(\frac{\min\{1, m_0\}}{p}-C_1M_p^p\bigg)- C_1M_q^qt^{q-p}-\lambda M_p^r\|h\|_{\frac{p}{p-r}}t^{r-p}. $

因为$ 1<r<p<q $, 故存在

$ \rho: = \bigg(\frac{\lambda(r-p)M_p^r\|h\|_{\frac{p}{p-r}}}{C_1M_q^q(p-q)}\bigg)^{\frac{1}{q-r}}>0, $

使得$ \max\limits_{t\in {{\Bbb R}}^{+}}\eta(t) = \eta(\rho) $.取

$ \lambda_0: = \bigg(\frac{\min\{1, m_0\}}{p}-C_1M_p^p)\bigg)^{\frac{q-r}{q-p}} \bigg(\frac{r-q}{p-q}M_p^r\|h\|_{\frac{p}{p-r}}\bigg)^{\frac{r-q}{q-p}} \bigg(\frac{(r-p)M_p^r\|h\|_{\frac{p}{p-r}}}{(p-q)C_1M_q^q}\bigg)^{\frac{p-r}{q-p}}, $

当$ \lambda<\lambda_0 $时, $ \eta(\rho)>0 $, 即对一切的$ u\in Z $, 存在$ \rho>0 $, 当$ \|u\| = \rho $时, 成立$ I(u)\geq \alpha: = \rho^p \eta(\rho)>0 $.

下面验证条件(A2).根据条件(f1), (f2)知, 存在正常数$ C_2, C_3 $, 使得

(3.3) $ \begin{equation} F(x, u)\geq C_2|u|^\mu-C_3|u|^p, \ \ \forall x\in {{{\Bbb R}} }^N, \ \ u\in {{\Bbb R}} . \end{equation} $

再根据条件(M2), 有

(3.4) $ \begin{equation} \overline{M}(\xi)\leq \overline{M}(1)\xi^{\theta}, \ \ \forall \xi\geq 1. \end{equation} $

在有限维空间$ E'\subset E $上定义泛函

$ I(u) = \frac{1}{p}\big(\overline{M}([u]_{s, p}^p)+\|u\|_{p.V}^p\big)-\int_{{{{\Bbb R}} }^N}F(x, u){\rm d}x -\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x. $

当$ [u]_{s, p}^p\geq 1 $时, 结合$ (3.3) $式和$ (3.4) $式可知

$ \begin{eqnarray*} I(u)&\leq & \frac{1}{p}\big(\overline{M}(1)[u]_{s, p}^{\theta p}+\|u\|_{p.V}^p\big)-C_2 \|u\|_{\mu}^\mu+C_3\|u\|_p^p-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ &\leq & \frac{\max\{1, \overline{M}(1)\}}{p}\|u\|^{\theta p}-C_2 \|u\|_{\mu}^\mu+C_3\|u\|_p^p-\frac{\lambda}{r}\|u\|_{L^r({{{\Bbb R}} }^N, h)}^r. \end{eqnarray*} $

因为有限维空间上所有的范数等价, 且$ \mu>\theta p\geq p>r>1 $, 故当$ [u]_{s, p}^p\geq 1 $, $ \|u\|\rightarrow \infty $时, $ I(u)\rightarrow -\infty $.即存在$ R>0 $, 使得当$ u\in E' $, $ \|u\|>R $时, $ I(u)<0 $.于是, 根据定理2.1可知方程(1.1)有无穷多解, 且对应的能量值趋于无穷大.定理1.1证毕.

定理1.2的证明   根据引理3.2, 对任意的$ c\in {{\Bbb R}} $, $ I $满足$ (PS)_c^{*} $条件.下面仅需要证明$ I $满足定理2.2的(B1)–(B3)条件.首先根据引理3.3, 当$ k\rightarrow \infty $时, $ \beta_\nu(k)\rightarrow 0 $, $ \nu\in [p, p_s^*) $.因此, 存在$ k_1>0 $, 当$ k>k_1 $时, $ \beta_p^p(k)\leq \frac{\min\{1, m_0\}}{2C_1} $.又因为$ 1<p<q<p_s^{*} $, 于是存在$ R\in (0, 1) $, 使得当$ R $充分小时, 对一切的$ u\in E $, $ \|u\|\leq R $, 成立

$ \frac{\min\{1, m_0\}}{4p}\|u\|^p\geq \frac{C_1}{q}M_q^q\|u\|^q. $

于是, 当$ u\in Z_k $, $ \|u\|\leq R $时, 有

$ \begin{eqnarray*} I(u)& = &\frac{1}{p}\big(\overline{M}([u]_{s, p}^p)+\|u\|_{p.V}^p\big)-\int_{{{{\Bbb R}} }^N}F(x, u){\rm d}x-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ &\geq & \frac{\min\{1, m_0\}}{p}\|u\|^p- \frac{C_1}{p}\int_{{{{\Bbb R}} }^N}|u|^p{\rm d}x- \frac{C_1}{q}\int_{{{{\Bbb R}} }^N}|u|^q{\rm d}x-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ &\geq & \frac{\min\{1, m_0\}}{p}\|u\|^p-\frac{C_1}{p}\beta_p^p(k)\|u\|^p- \frac{C_1}{q}M_q^q\|u\|^q-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ &\geq & \frac{\min\{1, m_0\}}{2p}\|u\|^p-\frac{C_1}{q}M_q^q\|u\|^q-\frac{\lambda }{r}\|h\|_{\frac{p}{p-r}}\beta_p^{r}(k)\|u\|^r\\ &\geq & \frac{\min\{1, m_0\}}{4p}\|u\|^p-\frac{\lambda }{r}\|h\|_{\frac{p}{p-r}}\beta_p^{r}(k)\|u\|^r, \end{eqnarray*} $

对任意的$ k>k_1 $, 令$ r_k = \big(\frac{4}{\min\{1, m_0\}}\lambda \|h\|_{\frac{p}{p-r}}\beta_p^r(k)\big)^{\frac{1}{p-r}} $.根据引理3.3, 当$ k\rightarrow \infty $时, $ r_k\rightarrow 0 $.故存在正整数$ k_0>k_1 $, 当$ k\geq k_0 $时, $ r_k\leq R $, 即对任意的$ k\geq k_0 $, $ u\in Z_k $, $ \|u\| = r_k $时, $ I(u)\geq 0 $.

其次, 取$ u\in Y_k $, $ \|u\| = \gamma_k $, $ 0<\gamma_k<\min\{1, r_k\} $, 结合中值定理和(3.3)式, 可得

$ \begin{eqnarray*} I(u) & = & \frac{1}{p}\big(\overline{M}([u]_{s, p}^p)+\|u\|_{p.V}^p\big)-\int_{{{{\Bbb R}} }^N}F(x, u){\rm d}x-\frac{\lambda}{r}\int_{{{{\Bbb R}} }^N}h(x)|u(x)|^{r}{\rm d}x\\ & \leq & \frac{\max\{1, \max\limits_{0<\tau<1}M(\tau)\}}{p}\|u\|^{ p}-C_2 \|u\|_{\mu}^\mu+C_3\|u\|_p^p-\frac{\lambda}{r}\|u\|_{L^r({{{\Bbb R}} }^N, h)}^r. \end{eqnarray*} $

因为有限维空间$ Y_k $上范数等价, 且$ \mu> p>r>1 $, 故当$ \gamma_k $充分小时, $ I(u)<0 $.

最后, 根据条件(B1), 当$ k\geq k_0 $, $ u\in Z_k $, $ \|u\|\leq r_k $时, 有

$ I(u)\geq -\frac{\lambda }{r}\|h\|_{\frac{p}{p-r}}\beta_p^{r}(k)\|u\|^r\geq- \frac{\lambda }{r}\|h\|_{\frac{p}{p-r}}\beta_p^{r}(k) r_k^r. $

因为当$ k\rightarrow \infty $时, $ \beta_p^r(k)\rightarrow 0 $, $ r_k\rightarrow 0 $, 故条件(B3)成立.根据定理2.2, 对任意的$ \lambda>0 $, 方程(1.1)有一列能量值趋于0的负能量解.定理1.2证毕.