# 含Φ-Laplace算子的拟线性椭圆型方程解的存在性Existence of Solutions for Quasilinear Elliptic Equation with Φ-Laplacian Operator

DOI: 10.12677/PM.2021.114069, PDF, HTML, XML, 下载: 15  浏览: 55

Abstract: The paper deals with the existence results of solutions to a quasilinear elliptic problem with Φ-Laplacian operator and potential well on ℝN. Nehari manifold and fibering maps are used to obtain the existence two nontrivial solutions.

1. 引言

$-{\Delta }_{\Phi }u+V\left(x\right)\varphi \left(|u|\right)u=f\left(x,u\right),\text{}u\in {W}^{1,\Phi }\left({ℝ}^{N}\right)$ (1)

$\left\{\begin{array}{l}-div\left(\varphi \left(|\nabla u|\right)\nabla u\right)=f\left(x,u\right),\text{}x\in \Omega ,\\ u=0,x\in \partial \Omega \text{ }\text{ }.\end{array}$ (2)

$\left({\varphi }_{1}\right)$ $t↦t\varphi \left(t\right)$$\left(0,+\infty \right)$ 是增函数；

$\left({\varphi }_{2}\right)$ $\underset{t\to 0}{\mathrm{lim}}t\varphi \left(t\right)=0$$\underset{t\to \infty }{\mathrm{lim}}t\varphi \left(t\right)=\infty$

$\left({\varphi }_{3}\right)$ 对任意 $t>0$，存在常数 $l,m\in \left(1,N\right)$，使得

$-10}{\mathrm{inf}}\frac{{\left(t\varphi \left(t\right)\right)}^{\prime \text{​}\prime }t}{{\left(t\varphi \left(t\right)\right)}^{\prime }}\le \underset{t>0}{\mathrm{sup}}\frac{{\left(t\varphi \left(t\right)\right)}^{\prime \text{​}\prime }t}{{\left(t\varphi \left(t\right)\right)}^{\prime }}=:m-2

$\left({\varphi }_{4}\right)$ 存在N-函数

$\Psi \left(t\right)={\int }_{0}^{t}\psi \left(s\right)\text{d}s$

$\left({a}_{1}\right)$ $10}{\mathrm{inf}}\frac{\psi \left(t\right)t}{\Psi \left(t\right)}\le \underset{t>0}{\mathrm{sup}}\frac{\psi \left(t\right)t}{\Psi \left(t\right)}=:{m}_{\Psi }<{l}^{*}=\frac{lN}{N-l}$

$\left({V}_{1}\right)$ $V\in C\left({ℝ}^{N}\right)$${V}_{0}=\underset{{ℝ}^{N}}{\mathrm{inf}}V>0$

$\left({V}_{2}\right)$ 对所有的 $M>0$$\mu \left({V}^{-1}\left(-\infty ,M\right]\right)<\infty$，其中 $\mu$${ℝ}^{N}$ 中的Lebesgue测度。

$\left({f}_{1}\right)$ 存在常数 $C>0$，使得

$|f\left(x,t\right)|\le C\left(1+\psi \left(t\right)\right)$$t\in ℝ$$x\in {ℝ}^{N}$

$\left({f}_{2}\right)$ $\underset{t\to 0}{\mathrm{lim}}\frac{f\left(x,t\right)}{\psi \left(t\right)}<\lambda$ 对几乎所有的 $x\in {ℝ}^{N}$ 一致成立；

$\left({f}_{3}\right)$ $t↦\frac{f\left(x,t\right)}{{|t|}^{m-2}t}$$ℝ\\left\{0\right\}$ 上单调递增；

$\left({f}_{4}\right)$ $\underset{|t|\to \infty }{\mathrm{lim}}\frac{f\left(x,t\right)}{{|t|}^{m-2}t}=+\infty$ 对几乎所有的 $x\in {ℝ}^{N}$ 一致成立。

$l-\text{2}\le \underset{t>0}{\mathrm{inf}}\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\le \underset{t>0}{\mathrm{sup}}\frac{{\varphi }^{\prime }\left(t\right)t}{\varphi \left(t\right)}\le m-2$$10}{\mathrm{inf}}\frac{\varphi \left(t\right){t}^{2}}{\Phi \left(t\right)}\le \underset{t>0}{\mathrm{sup}}\frac{\varphi \left(t\right){t}^{2}}{\Phi \left(t\right)}=:m

$\Phi \left(t\right)={|t|}^{a}\mathrm{log}\left(1+|t|\right)$${\psi }_{1}\left(t\right)=q{t}^{q-1}\mathrm{log}\left(1+t\right)+\frac{{t}^{q}}{t+1}$$f\left(x,t\right)=\left\{\begin{array}{l}{\psi }_{\text{2}}\left(t\right),0

2. 预备知识和基本引理

${L}^{\Phi }\left(\Omega \right)=\left\{u:\Omega \to ℝ是可测的,{\int }_{\Omega }\Phi \left(|u\left(x\right)|\right)\text{d}x<\infty \right\}$，在Luxemburg范数

${‖u‖}_{\Phi }=\mathrm{inf}\left\{k>0|{\int }_{\Omega }\Phi \left(\frac{|u\left(x\right)|}{k}\right)\text{d}x\le 1\right\}$

${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)=\left\{u\in {W}^{1,\Phi }\left({ℝ}^{N}\right)|{\int }_{{ℝ}^{N}}V\left(x\right)\Phi \left(|u\left(x\right)|\right)<\infty \right\}$

${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$ 上定义范数 $‖u‖={‖\nabla u‖}_{\Phi }\text{+}{‖u‖}_{\Phi ,V}$，其中

${‖u‖}_{\Phi ,V}=\mathrm{inf}\left\{k>0|{\int }_{{ℝ}^{N}}V\left(x\right)\Phi \left(\frac{|u\left(x\right)|}{k}\right)\text{d}x\le 1\right\}$

${\xi }_{1}\left(t\right)=\mathrm{min}\left\{{t}^{\mathcal{l}},{t}^{m}\right\}$${\xi }_{2}\left(t\right)=\mathrm{max}\left\{{t}^{\mathcal{l}},{t}^{m}\right\}$

${\xi }_{1}\left(t\right)\Phi \left(\rho \right)\le \Phi \left(\rho t\right)\le {\xi }_{2}\left(t\right)\Phi \left(\rho \right)$${\xi }_{1}\left({‖u‖}_{\Phi }\right)\le {\int }_{{ℝ}^{N}}\Phi \left(|u\left(x\right)|\right)\text{d}x\le {\xi }_{2}\left({‖u‖}_{\Phi }\right)$

${\xi }_{1}\left({‖u‖}_{\Phi ,V}\right)\le {\int }_{{ℝ}^{N}}V\left(x\right)\Phi \left(|u\left(x\right)|\right)\text{d}x\le {\xi }_{2}\left({‖u‖}_{\Phi ,V}\right)$

${\xi }_{1}\left(t\right)\Psi \left(\rho \right)\le \Psi \left(\rho t\right)\le {\xi }_{2}\left(t\right)\Psi \left(\rho \right)$${\xi }_{1}\left({‖u‖}_{\Psi }\right)\le {\int }_{{ℝ}^{N}}\Psi \left(|u\left(x\right)|\right)\text{d}x\le {\xi }_{2}\left({‖u‖}_{\Psi }\right)$

$\stackrel{¯}{\underset{t\to 0}{\mathrm{lim}}}\frac{\Psi \left(t\right)}{\Phi \left(t\right)}<+\infty$$\stackrel{¯}{\underset{|t|\to +\infty }{\mathrm{lim}}}\frac{\Psi \left(t\right)}{{\Phi }_{*}\left(t\right)}=0$

$I\left(u\right)={\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}F\left(x,u\right)\text{d}x$$u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$(3)

$〈{I}^{\prime }\left(u\right),\phi 〉={\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla u|\right)\nabla u\nabla \phi +V\left(x\right)\varphi \left(|u|\right)u\phi \right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,u\right)\phi \text{d}x$

$\mathcal{M}=\left\{u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)\\left\{0\right\}:〈{I}^{\prime }\left(u\right),u〉=0\right\}$

${h}_{u}\left(t\right)={\int }_{{ℝ}^{N}}\left(\Phi \left(t|\nabla u|\right)+V\left(x\right)\Phi \left(t|u|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}F\left(x,tu\right)\text{d}x$

$\left({\varphi }_{1}\right)\text{-}\left({\varphi }_{3}\right)$ 条件可得纤维映射 ${h}_{u}\left(t\right)\in {C}^{2}$，且

${{h}^{\prime }}_{u}\left(t\right)={\int }_{{ℝ}^{N}}t\left(\varphi \left(t|\nabla u|\right){|\nabla u|}^{2}+V\left(x\right)\varphi \left(t|u|\right){u}^{2}\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)u\text{d}x$

$\begin{array}{c}{{h}^{″}}_{u}\left(t\right)={\int }_{{ℝ}^{N}}\left(t{\varphi }^{\prime }\left(t|\nabla u|\right){|\nabla u|}^{3}+\varphi \left(t|\nabla u|\right){|\nabla u|}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{ℝ}^{N}}\left(tV\left(x\right){\varphi }^{\prime }\left(t|u|\right){|u|}^{3}+V\left(x\right)\varphi \left(t|u|\right){u}^{2}\right)\text{d}x-{\int }_{{ℝ}^{N}}{f}^{\prime }\left(x,tu\right){u}^{2}\text{d}x\end{array}$

3. 定理1.1的证明

(1) ${\int }_{{ℝ}^{N}}f\left(x,{u}_{n}\right){u}_{n}\text{d}x\to {\int }_{{ℝ}^{N}}f\left(x,u\right)u\text{d}x$ ；(2) ${\int }_{{ℝ}^{N}}F\left(x,{u}_{n}\right)\text{d}x\to {\int }_{{ℝ}^{N}}F\left(x,u\right)\text{d}x$

${u}_{n}⇀u$ (在 ${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$ 中)。

${u}_{n}\to u$ (在 ${L}^{\Psi }\left({ℝ}^{N}\right)$ 中)。

${u}_{n}\to u$ a.e. $x\in {ℝ}^{N}$$|{u}_{n}|\le h$ a.e. $x\in {ℝ}^{N}$

$\left({f}_{1}\right)$$\left({a}_{1}\right)$ 可得

$|f\left(x,{u}_{n}\right){u}_{n}|\le C|{u}_{n}|+C\psi \left({u}_{n}\right)|{u}_{n}|\le Ch+C\psi \left(h\right)h\le Ch+C{m}_{\Psi }\Psi \left(h\right)\in {L}^{1}\left({ℝ}^{N}\right)$

$\underset{n\to \infty }{\mathrm{lim}}{\int }_{{ℝ}^{N}}f\left(x,{u}_{n}\right){u}_{n}\text{d}x={\int }_{{ℝ}^{N}}f\left(x,u\right)u\text{d}x$

$\underset{t\to 0}{\mathrm{lim}}\frac{F\left(x,t\right)}{\Psi \left(t\right)}=\underset{t\to 0}{\mathrm{lim}}\frac{f\left(x,t\right)}{\psi \left(t\right)}<\lambda$

$\frac{|F\left(x,t\right)|}{\Psi \left(t\right)}<\lambda -\epsilon$$x\in {ℝ}^{N}$$|t|<\delta$

$|F\left(x,{u}_{n}\right)|<\left(\lambda -\epsilon \right)|\Psi \left({u}_{n}\right)|\le \left(\lambda -\epsilon \right)\Psi \left(h\right)\in {L}^{1}\left({ℝ}^{N}\right)$

$|f\left(x,t\right)|\le \left(\lambda -\epsilon \right)|\psi \left(t\right)|+{C}_{\epsilon }\psi \left(t\right)$$t\in ℝ$

$|F\left(x,t\right)|\le \left(\lambda -\epsilon \right)\Psi \left(t\right)+{C}_{\epsilon }\Psi \left(t\right)$$t\in ℝ$(4)

$\left({f}_{2}\right)$$\left({a}_{1}\right)$，可得

$\underset{t\to 0}{\mathrm{lim}\mathrm{sup}}\frac{f\left(x,t\right)t}{\Psi \left(t\right)}<\lambda {m}_{\Psi }$(5)

$|f\left(x,t\right)t|\le \left(\lambda {m}_{\Psi }-\epsilon \right)\Psi \left(t\right)+{C}_{\epsilon }\Psi \left(t\right)$$t\in ℝ$(6)

${\int }_{{ℝ}^{N}}f\left(x,u\right)u\text{d}x\le \left(\lambda {m}_{\Psi }-\epsilon \right){\int }_{{ℝ}^{N}}\Psi \left(u\right)\text{d}x+{C}_{\epsilon }{\int }_{{ℝ}^{N}}\Psi \left(u\right)\text{d}x$

${\int }_{{ℝ}^{N}}F\left(x,u\right)\text{d}x\le \left(\lambda -\epsilon \right){\int }_{{ℝ}^{N}}\Psi \left(u\right)\text{d}x+{C}_{\epsilon }{\int }_{{ℝ}^{N}}\Psi \left(u\right)\text{d}x$

${\int }_{{ℝ}^{N}}f\left(x,u\right)u\text{d}x\le \left(\lambda {m}_{\Psi }-\epsilon \right)\mathrm{max}\left\{{‖u‖}^{{l}_{\Psi }},{‖u‖}^{{m}_{\Psi }}\right\}+{C}_{\epsilon }\mathrm{max}\left\{{‖u‖}^{{l}_{\Psi }},{‖u‖}^{{m}_{\Psi }}\right\}$(7)

${\int }_{{ℝ}^{N}}F\left(x,u\right)\text{d}x\le \left(\lambda -\epsilon \right)\mathrm{max}\left\{{‖u‖}^{{l}_{\Psi }},{‖u‖}^{{m}_{\Psi }}\right\}+{C}_{\epsilon }\mathrm{max}\left\{{‖u‖}^{{l}_{\Psi }},{‖u‖}^{{m}_{\Psi }}\right\}$(8)

(1) $\underset{t\to 0}{\mathrm{lim}}\frac{{{h}^{\prime }}_{u}\left(t\right)}{{t}^{m-1}}>0$$\underset{t\to \infty }{\mathrm{lim}}\frac{{{h}^{\prime }}_{u}\left(t\right)}{{t}^{m-1}}=-\infty$ ；(2) $\underset{t\to 0}{\mathrm{lim}}\frac{{h}_{u}\left(t\right)}{{t}^{m}}>0$$\underset{t\to \infty }{\mathrm{lim}}\frac{{h}_{u}\left(t\right)}{{t}^{m}}=-\infty$

${{h}^{\prime }}_{u}\left(t\right)={\int }_{{ℝ}^{N}}t\left(\varphi \left(t|\nabla u|\right){|\nabla u|}^{2}+V\left(x\right)\varphi \left(t|u|\right){u}^{2}\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)u\text{d}x$

$\begin{array}{c}t{{h}^{\prime }}_{u}\left(t\right)={\int }_{{ℝ}^{N}}\left(\varphi \left(t|\nabla u|\right){|t\nabla u|}^{2}+V\left(x\right)\varphi \left(t|u|\right){|tu|}^{2}\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)tu\text{d}x\\ \ge l{\int }_{{ℝ}^{N}}\left(\Phi \left(t|\nabla u|\right)+V\left(x\right)\Phi \left(t|u|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)tu\text{d}x\\ \ge l{t}^{m}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-\left(\lambda {m}_{\Psi }-\epsilon +{C}_{\epsilon }\right)\mathrm{max}\left\{{‖tu‖}^{{l}_{\Psi }},{‖tu‖}^{{m}_{\Psi }}\right\}\end{array}$

$\frac{{{h}^{\prime }}_{u}\left(t\right)}{{t}^{m-1}}\ge l{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-\left(\lambda {m}_{\Psi }-\epsilon +{C}_{\epsilon }\right)\frac{\mathrm{max}\left\{{‖tu‖}^{{l}_{\Psi }},{‖tu‖}^{{m}_{\Psi }}\right\}}{{t}^{m}}$

$\begin{array}{c}t{{h}^{\prime }}_{u}\left(t\right)={\int }_{{ℝ}^{N}}\left(\varphi \left(t|\nabla u|\right){|t\nabla u|}^{2}+V\left(x\right)\varphi \left(|tu|\right){|tu|}^{2}\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)tu\text{d}x\\ \le m{t}^{m}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}f\left(x,tu\right)tu\text{d}x\end{array}$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\frac{f\left(x,tu\right)tu}{{t}^{m}}\text{d}x=\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\frac{f\left(x,tu\right)}{{|tu|}^{m-2}tu}{|u|}^{m}\text{d}x=+\infty$

$\underset{t\to \infty }{\mathrm{lim}}\frac{{{h}^{\prime }}_{u}\left(t\right)}{{t}^{m-1}}\le m{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\frac{f\left(x,tu\right)tu}{{t}^{m}}\text{d}x=-\infty$

$u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)↦{\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla u|\right){|\nabla u|}^{2}+V\left(x\right)\varphi \left(|u|\right){u}^{2}\right)\text{d}x$

$u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)↦{\int }_{{ℝ}^{N}}\left(m\Phi \left(|\nabla u|\right)-\varphi \left(|\nabla u|\right){|\nabla u|}^{2}+V\left(x\right)\left(m\Phi \left(|u|\right)-\varphi \left(|u|\right){u}^{2}\right)\right)\text{d}x$

${t}_{\mathrm{max}}\left(u\right)>0$，使得 ${t}_{\mathrm{max}}\left(u\right)u\in \mathcal{M}$，并且对任意 $u\in \mathcal{M}$，有 $I\left(u\right)>0$

$\frac{\text{d}}{\text{d}t}\left(\frac{\varphi \left(t|\nabla u|\right)\nabla \left(tu\right)\nabla u}{{t}^{m-1}}\right)=\frac{{|\nabla u|}^{2}\left({\varphi }^{\prime }\left(t|\nabla u|\right)|\nabla \left(tu\right)|-\left(m-2\right)\varphi \left(t|\nabla u|\right)\right)}{{t}^{m-1}}\le 0$(9)

$\frac{\text{d}}{\text{d}t}\left(\frac{V\left(x\right)\varphi \left(t|u|\right)t{u}^{2}}{{t}^{m-1}}\right)=\frac{{u}^{2}\left({\varphi }^{\prime }\left(t|u|\right)|tu|-\left(m-2\right)V\left(x\right)\varphi \left(t|u|\right)\right)}{{t}^{m-1}}\le 0$(10)

${{h}^{\prime }}_{u}\left(t\right)=〈{I}^{\prime }\left(tu\right),u〉$，结合(3.6)、(3.7)和 $\left({f}_{3}\right)$，对任意 $t>0$$u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)\\left\{0\right\}$

$\frac{\text{d}}{\text{d}t}\left(\frac{{{h}^{\prime }}_{u}\left(t\right)}{{t}^{m-1}}\right)\le -{\int }_{{ℝ}^{N}}\frac{\text{d}}{\text{d}t}\left(\frac{f\left(x,tu\right)}{{|tu|}^{m-2}tu}\right){|u|}^{m}\text{d}x<0$(11)

${t}_{\mathrm{max}}\left(u\right)u\in \mathcal{M}$，由(11)式可知，对任意 $u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)\\left\{0\right\}$$t>0$，有 ${{h}^{″}}_{u}\left(t\right)<0$。根据引理3.2(1)知 $h\left({t}_{\mathrm{max}}\left(u\right)\right)>0$，故 $I\left({t}_{\mathrm{max}}\left(u\right)u\right)>0$。因为 $u\in \mathcal{M}$ 当且仅当 ${t}_{\mathrm{max}}\left(u\right)=1$，故对任意 $u\in \mathcal{M}$，有 $I\left(u\right)>0$。证毕。

$J\left(u\right)={\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla u|\right){|\nabla u|}^{2}+V\left(x\right)\varphi \left(|u|\right){u}^{2}\right)\text{d}x$

$J\left(u\right)$${C}^{1}$ 的，且

$〈{J}^{\prime }\left(u\right),v〉={\int }_{{ℝ}^{N}}\left(2\varphi \left(|\nabla u|\right)+{\varphi }^{\prime }\left(|\nabla u|\right)|\nabla u|\right)\nabla u\nabla v\text{d}x+{\int }_{{ℝ}^{N}}V\left(x\right)\left(2\varphi \left(|u|\right)+{\varphi }^{\prime }\left(|u|\right)|u|\right)uv\text{d}x$$u,v\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$

${G}^{\prime }\left(u\right)={I}^{″}\left(u\right)\cdot \left(u,u\right)+〈{I}^{\prime }\left(u\right),u〉={{h}^{″}}_{u}\left(1\right)<0$$\forall u\in \mathcal{M}$

$\left\{\begin{array}{l}\mathrm{min}I\left(u\right)\\ h\left(u\right)=0\end{array}$

${I}^{\prime }\left({u}_{0}\right)={\mu }_{1}{G}^{\prime }\left({u}_{0}\right)$

$〈{I}^{\prime }\left({u}_{0}\right),{u}_{0}〉={\mu }_{1}〈{G}^{\prime }\left({u}_{0}\right),{u}_{0}〉=0$

$〈{G}^{\prime }\left({u}_{0}\right),{u}_{0}〉={{h}^{″}}_{u}\left(1\right)<0$$\forall {u}_{0}\in M$

$\begin{array}{l}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x\\ \le \frac{1}{l}{\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla {u}_{n}|\right)\nabla {u}_{n}^{2}+V\left(x\right)\varphi \left(|{u}_{n}|\right){u}_{n}^{2}\right)\text{d}x\\ =\frac{1}{l}{\int }_{{ℝ}^{N}}f\left(x,{u}_{n}\right){u}_{n}\text{d}x\\ \le \frac{1}{l}\left(\lambda {m}_{\Psi }-\epsilon \right){\int }_{{ℝ}^{N}}\Psi \left(|{u}_{n}|\right)\text{d}x+\frac{{C}_{\epsilon }}{l}{\int }_{{ℝ}^{N}}\Psi \left(|{u}_{n}|\right)\text{d}x\\ \le \frac{1}{l}\left(\lambda {m}_{\Psi }-\epsilon \right)\mathrm{max}\left\{{‖{u}_{n}‖}^{{l}_{\Psi }},{‖{u}_{n}‖}^{{m}_{\Psi }}\right\}+\frac{{C}_{\epsilon }}{l}\mathrm{max}\left\{{‖{u}_{n}‖}^{{l}_{\Psi }},{‖{u}_{n}‖}^{{m}_{\Psi }}\right\}\\ ={C}_{1}\mathrm{max}\left\{{‖{u}_{n}‖}^{{l}_{\Psi }},{‖{u}_{n}‖}^{{m}_{\Psi }}\right\}+{C}_{2}\mathrm{max}\left\{{‖{u}_{n}‖}^{{l}_{\Psi }},{‖{u}_{n}‖}^{{m}_{\Psi }}\right\}\end{array}$

$\begin{array}{l}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x\\ \ge \mathrm{min}\left\{{‖\nabla {u}_{n}‖}_{\Phi }^{l},{‖\nabla {u}_{n}‖}_{\Phi }^{m}\right\}\ge {\left(\frac{1}{2}\right)}^{m}\mathrm{min}\left\{{‖{u}_{n}‖}^{l},{‖{u}_{n}‖}^{m}\right\}={\left(\frac{1}{2}\right)}^{m}{‖{u}_{n}‖}^{m}\end{array}$

${\left(\frac{1}{2}\right)}^{m}{‖{u}_{n}‖}^{m}\le {\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x\le C{‖{u}_{n}‖}^{{l}_{\Psi }}+{C}_{2}{‖{u}_{n}‖}^{{l}_{\Psi }}\le {C}_{3}{‖{u}_{n}‖}^{{l}_{\Psi }}$

${C}_{\mathcal{M}}=\underset{u\in \mathcal{M}}{\mathrm{inf}}I\left(u\right)$，首先证明I的任何极小化序列在 ${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$ 上是有界的。

$\underset{n\to \infty }{\mathrm{lim}}‖{u}_{n}‖=\infty$，且 $‖{u}_{n}‖>1$$\forall n\in N$

$\frac{I\left({u}_{n}\right)}{{‖{u}_{n}‖}^{m}}={ο}_{n}\left(1\right)$

${v}_{n}=\frac{{u}_{n}}{‖{u}_{n}‖}$$n=1,2,\cdots$，则 $\left\{{v}_{n}\right\}\subset {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$，且 $‖{v}_{n}‖=1$$\forall n\in N$。因此存在 ${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$ 中的元素v，使得 ${v}_{n}⇀v$ (在 ${W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$ 中)。

$I\left({u}_{n}\right)=\underset{t>0}{\mathrm{max}}I\left(t{u}_{n}\right)$$\forall n\in N$

${C}_{M}+{ο}_{n}\left(1\right)\ge I\left(z{v}_{n}\right)={\int }_{{ℝ}^{N}}\left(\Phi \left(z|\nabla {v}_{n}|\right)+V\left(x\right)\Phi \left(z|{v}_{n}|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}F\left(x,z{v}_{n}\right)\text{d}x$

$\begin{array}{c}{C}_{M}+{ο}_{n}\left(1\right)=I\left({u}_{n}\right)\ge {\int }_{{ℝ}^{N}}\left(\Phi \left(z|\nabla {v}_{n}|\right)+V\left(x\right)\Phi \left(z|{v}_{n}|\right)\right)\text{d}x+{ο}_{n}\left(1\right)\\ \ge \mathrm{min}\left\{{‖z\nabla {v}_{n}‖}_{\Phi }^{l},{‖z\nabla {v}_{n}‖}_{\Phi }^{m}\right\}+\mathrm{min}\left\{{‖z{v}_{n}‖}_{\Phi ,V}^{l},{‖z{v}_{n}‖}_{\Phi ,V}^{m}\right\}+{ο}_{n}\left(1\right)\\ \ge \mathrm{min}\left\{{‖z\nabla {v}_{n}‖}_{\Phi }^{l},{‖z\nabla {v}_{n}‖}_{\Phi }^{m}\right\}+{ο}_{n}\left(1\right)\\ \ge \mathrm{min}\left\{{‖\frac{1}{2}z{v}_{n}‖}^{l},{‖\frac{1}{2}z{v}_{n}‖}^{m}\right\}+{ο}_{n}\left( 1 \right)\end{array}$

$n\to \infty$，由引理2.1，

$\begin{array}{c}{\int }_{{ℝ}^{N}}\frac{F\left(x,{u}_{n}\right)}{{‖{u}_{n}‖}^{m}}\text{d}x=\frac{1}{{‖{u}_{n}‖}^{m}}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x+{ο}_{n}\left(1\right)\\ \le {\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {v}_{n}|\right)+V\left(x\right)\Phi \left(|{v}_{n}|\right)\right)\text{d}x+{ο}_{n}\left(1\right)\\ \le \mathrm{max}\left\{{‖\nabla {v}_{n}‖}_{\Phi }^{l},{‖\nabla {v}_{n}‖}_{\Phi }^{m}\right\}+\mathrm{max}\left\{{‖{v}_{n}‖}_{\Phi ,V}^{l},{‖{v}_{n}‖}_{\Phi ,V}^{m}\right\}+{ο}_{n}\left(1\right)\\ ={‖\nabla {v}_{n}‖}_{\Phi }^{l}+{‖{v}_{n}‖}_{\Phi ,V}^{l}+{ο}_{n}\left(1\right)\\ \le ‖{v}_{n}‖+{ο}_{n}\left(1\right)=1+{ο}_{n}\left( 1 \right)\end{array}$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}{\int }_{{ℝ}^{N}}\frac{F\left(x,{u}_{n}\right)}{{‖{u}_{n}‖}^{m}}\text{d}x\le 1$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\frac{F\left(x,{u}_{n}\right)}{{‖{u}_{n}‖}^{m}}\text{d}x\ge {\int }_{{ℝ}^{N}}\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{F\left(x,{u}_{n}\right)}{{‖{u}_{n}‖}^{m}}\text{d}x={\int }_{{ℝ}^{N}}\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{F\left(x,{u}_{n}\right)}{{|{u}_{n}|}^{m}}{|{v}_{n}|}^{m}\text{d}x=+\infty$

$\begin{array}{c}0\le {\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x\\ \le \frac{1}{l}{\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla {u}_{n}|\right){|\nabla {u}_{n}|}^{2}+V\left(x\right)\varphi \left(|{u}_{n}|\right){u}_{n}^{2}\right)\text{d}x\\ =\frac{1}{l}{\int }_{{ℝ}^{N}}f\left(x,{u}_{n}\right){u}_{n}\text{d}x\end{array}$ (12)

$〈{I}^{\prime }\left(u\right),u〉\le \underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}〈{I}^{\prime }\left({u}_{n}\right),{u}_{n}〉=0$

$\begin{array}{c}{C}_{\mathcal{M}}\le I\left(tu\right)=I\left(tu\right)-\frac{1}{m}〈{I}^{\prime }\left(tu\right),tu〉\\ ={\int }_{{ℝ}^{N}}\left(\Phi \left(t|\nabla u|\right)-\frac{1}{m}\varphi \left(t|\nabla u|\right){t}^{2}{|\nabla u|}^{2}+V\left(x\right)\Phi \left(t|u|\right)-\frac{1}{m}V\left(x\right)\varphi \left(t|u|\right){t}^{2}{u}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{ℝ}^{N}}\left(\frac{1}{m}f\left(x,tu\right)tu-F\left(x,tu\right)\right)\text{d}x\end{array}$ (13)

$\left({f}_{1}\right)$ 可得

$\frac{\text{d}}{\text{d}t}\left\{\frac{1}{m}f\left(x,t\right)t-F\left(x,t\right)\right\}=\frac{{t}^{m}}{m}\cdot \frac{\text{d}}{\text{d}t}\left\{\frac{f\left(x,t\right)}{{t}^{m-1}}\right\}>0$

$t↦\Phi \left(t|\nabla u|\right)-\frac{1}{m}\varphi \left(t|\nabla u|\right){t}^{2}{|\nabla u|}^{2}+V\left(x\right)\Phi \left(t|u|\right)-\frac{1}{m}V\left(x\right)\varphi \left(t|u|\right){t}^{2}{u}^{2}$

$\left(0,\infty \right)$ 上递增。由(12)和(13)可得

$\begin{array}{c}{C}_{M}<{\int }_{{R}^{N}}\left(\Phi \left(|\nabla u|\right)-\frac{1}{m}\varphi \left(|\nabla u|\right){|\nabla u|}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{R}^{N}}\left(V\left(x\right)\Phi \left(|u|\right)-\frac{1}{m}V\left(x\right)\varphi \left(|u|\right){u}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{ℝ}^{N}}\left(\frac{1}{m}f\left(x,u\right)u-F\left(x,u\right)\right)\text{d}x\end{array}$

$\begin{array}{c}{C}_{M}<\underset{n\to \infty }{\mathrm{lim}}{\int }_{{R}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)-\frac{1}{m}\varphi \left(|\nabla {u}_{n}|\right){|\nabla {u}_{n}|}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{R}^{N}}\left(V\left(x\right)\Phi \left(|{u}_{n}|\right)-\frac{1}{m}V\left(x\right)\varphi \left(|{u}_{n}|\right){u}_{n}^{2}\right)\text{d}x\\ \text{\hspace{0.17em}}+{\int }_{{ℝ}^{N}}\left(\frac{1}{m}f\left(x,{u}_{n}\right){u}_{n}-F\left(x,{u}_{n}\right)\right)\text{d}x\\ =\underset{n\to \infty }{\mathrm{lim}}\left(I\left({u}_{n}\right)-\frac{1}{m}{I}^{\prime }\left({u}_{n}\right){u}_{n}\right)={C}_{M}\end{array}$

$\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}-\nabla u|\right)+V\left(x\right)\Phi \left(|{u}_{n}-u|\right)\right)\text{d}x\ge {\delta }_{1}>0$(14)

$\begin{array}{l}\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}{\int }_{{ℝ}^{N}}\left(\left(\Phi \left(|\nabla {u}_{n}|\right)-\Phi \left(|\nabla {u}_{n}-\nabla u|\right)\right)+V\left(x\right)\left(\Phi \left(|{u}_{n}|\right)-\Phi \left(|{u}_{n}-u|\right)\right)\right)\text{d}x\\ ={\int }_{{ℝ}^{N}}\left(\Phi \left(|u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x\end{array}$(15)

$\begin{array}{c}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x\le \underset{n\to \infty }{\mathrm{lim}}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x-{\delta }_{1}\\ <\underset{n\to \infty }{\mathrm{lim}}{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x\end{array}$

${C}_{\mathcal{M}}=\underset{n\to \infty }{\mathrm{lim}}I\left({u}_{n}\right)=\underset{n\to \infty }{\mathrm{lim}}\left\{{\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{n}|\right)+V\left(x\right)\Phi \left(|{u}_{n}|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}F\left(x,{u}_{n}\right)\text{d}x\right\}>I\left(u\right)$

${C}_{\mathcal{M}}=I\left(u\right)=\underset{u\in \mathcal{M}}{\mathrm{min}}I\left(u\right)>0$

4. 定理1.2的证明

${f}_{+}\left(x,t\right)=\left\{\begin{array}{l}f\left(x,t\right),t\ge 0,\\ 0,t<0,\end{array}$ ${f}_{-}\left(x,t\right)=\left\{\begin{array}{l}f\left(x,t\right),t\le 0,\\ 0,t>0,\end{array}$

${I}_{±}\left(u\right)={\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla u|\right)+V\left(x\right)\Phi \left(|u|\right)\right)\text{d}x-{\int }_{{ℝ}^{N}}{F}_{±}\left(x,u\right)\text{d}x$$u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)$

${\mathcal{M}}^{+}=\left\{u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)\\left\{0\right\}:〈{{I}^{\prime }}_{+}\left(u\right),u〉=0\right\}$

${\mathcal{M}}^{-}=\left\{u\in {W}_{V}^{1,\Phi }\left({ℝ}^{N}\right)\\left\{0\right\}:〈{{I}^{\prime }}_{-}\left(u\right),u〉=0\right\}$

${I}_{+}\left({u}_{1}\right)={c}^{+}>0$${I}_{-}\left({u}_{2}\right)={c}^{-}>0$

$\begin{array}{c}0\le {\int }_{{ℝ}^{N}}\left(\Phi \left(|\nabla {u}_{1}^{-}|\right)+V\left(x\right)\Phi \left(|{u}_{1}^{-}|\right)\right)\text{d}x\\ \le \frac{1}{l}{\int }_{{ℝ}^{N}}\left(\varphi \left(|\nabla {u}_{1}^{-}|\right){|\nabla {u}_{1}^{-}|}^{2}+V\left(x\right)\varphi \left(|{u}_{1}^{-}|\right){u}_{1}^{-}{}^{2}\right)\text{d}x\\ =\frac{1}{l}{\int }_{{ℝ}^{N}}f\left(x,{u}_{1}\right){u}_{1}^{-}\text{d}x=0\end{array}$ (16)

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