﻿ 一类含p-Laplace算子的奇异拟线性问题解的多重性

# 一类含p-Laplace算子的奇异拟线性问题解的多重性Multiplicity of Solutions of a Class of Singular Quasilinear Problems Involving p-Laplacian

Abstract: In this paper we combine sub-supersolution technique and minimax methods to study the existence and multiplicity of solutions for a class of singular quasilinear elliptic equations. Firstly, by suitable hypotheses on the nonlinearity term and singular term, we obtain the existence of nontrivial solutions. Furthermore, by strengthening the hypotheses and applying the Mountain Pass Theorem, we show the existence of a second solution.

1. 引言

$\left\{\begin{array}{l}-{\Delta }_{p}u-u{\Delta }_{p}\left({u}^{2}\right)=b\left(x\right){u}^{-\gamma }+f\left(x,u\right),\text{}x\in \Omega ,\\ u>0,\text{}x\in \Omega \text{,}u=0,\text{}x\in \partial \Omega ,\text{}\end{array}$ (1.1)

(b1)存在函数 ${e}_{0}\in {C}_{0}^{1}\left(\stackrel{¯}{\Omega }\right)$${e}_{0}\ge 0$$q>N$，使得 $b{e}_{0}^{-\gamma }\in {L}^{q}\left(\Omega \right)$

(f1)对几乎处处 $x\in \Omega$ 和任意 $s\in \left[0,\delta \right]$，存在常数 $\delta ,k>0$，使得

$-kb\left(x\right)\le f\left(x,s\right)$.

(f2)对几乎处处 $x\in \Omega$ 和任意 $s\in ℝ$，存在 ${F}_{0}>0$$r>2p$$0\le {a}_{1}\left(x\right)\in {L}^{\infty }\left(\Omega \right)$${a}_{1}\left(x\right)$ 不恒为0，使得

$|f\left(x,s\right)|\le {F}_{0}\left(1+{a}_{1}\left(x\right){|s|}^{r-1}\right).$

(f3)对几乎处处 $x\in \Omega$ 和任意 $s\in \left[{s}_{0},\infty \right)$，存在 ${s}_{0}>0$$\theta >p$，使得

$0<2\theta F\left(x,s\right)\le sf\left(x,s\right),$

$u\in {W}_{0}^{1,p}\left(\Omega \right)\cap {L}^{\infty }\left(\Omega \right)$$u>0$，成立

${\int }_{\Omega }\left(1+{2}^{p-1}{|u|}^{p}\right){|\nabla u|}^{p-2}\nabla u\nabla \phi +{2}^{p-1}{\int }_{\Omega }{|\nabla u|}^{p}{|u|}^{p-2}u\phi ={\int }_{\Omega }h\left(x,u\right)\phi ,\text{}\forall \phi \in {C}_{0}^{\infty }\left(\Omega \right),$ (1.2)

$J\left(u\right)=\frac{1}{p}{\int }_{\Omega }\left(1+{2}^{p-1}{|u|}^{p}\right){|\nabla u|}^{p}-{\int }_{\Omega }H\left(x,u\right),$

$\left\{\begin{array}{l}g\left(s\right)=-g\left(-s\right)\text{}s\in \left(-\infty ,0\right],\\ {g}^{\prime }\left(s\right)=\frac{1}{{\left(1+{2}^{p-1}{|g\left(s\right)|}^{p}\right)}^{1/p}}\text{}s\in \left[0,+\infty \right).\end{array}$ (1.3)

$J\left(g\left(v\right)\right)=\frac{1}{p}{\int }_{\Omega }{|\nabla v|}^{p}-{\int }_{\Omega }H\left(x,g\left(v\right)\right),$ (1.4)

$\left\{\begin{array}{l}-{\Delta }_{p}v=b\left(x\right){\left(g\left(v\right)\right)}^{-\gamma }{g}^{\prime }\left(v\right)+f\left(x,g\left(v\right)\right){g}^{\prime }\left(v\right)\text{}x\in \Omega ,\\ v>0\text{}x\in \Omega \text{,}v=0\text{}x\in \partial \Omega .\text{}\end{array}$ (1.5)

2. 预备知识和基本引理

${L}^{P}\left(\Omega \right)$ 是Lebesgue空间，并定义范数 ${‖u‖}_{p}={\left({\int }_{\Omega }{|u|}^{p}\text{d}x\right)}^{\frac{1}{p}},\text{\hspace{0.17em}}1\le p<\infty$${‖u‖}_{\infty }=\underset{x\in \Omega }{\text{esssup}}|u|,\text{\hspace{0.17em}}p=\infty$

${W}_{0}^{1,p}\left(\Omega \right)$ 是Sobolev空间，并定义范数 $‖u‖={\left({\int }_{\Omega }{|\nabla u|}^{p}\text{d}x\right)}^{\frac{1}{p}}$

1) 函数g唯一确定的，且是 ${C}^{2}$ 和可逆的；

2) 对所有 $s\in ℝ$$|{g}^{\prime }\left(s\right)|\le 1$

3) 对所有 $s\in ℝ$$|g\left(s\right)|\le |s|$

4) 当 $s\to 0$$g\left(s\right)/s\to 1$

5) 对所有 $s\in ℝ$$|g\left(s\right)|\le {2}^{\frac{1}{2p}}{|s|}^{\frac{1}{2}}$

6) 对所有 $s\ge 0$$g\left(s\right)/2\le s{g}^{\prime }\left(s\right)\le g\left(s\right)$

7) 当 $s\to \text{+}\infty$$g\left(s\right)/\sqrt{s}\to a>0$

8) 存在一个正常数C，使得

$|g\left(s\right)|\ge \left\{\begin{array}{l}C|s|,\text{}|s|\le 1,\\ C{|s|}^{\frac{1}{2}},\text{}|s|\ge 1;\end{array}$

9) 对所有 $s\ge 1$$t\ge 0$${g}^{2}\left(st\right)\ge s{g}^{2}\left(t\right)$

10) 对所有 $s\in ℝ$$|g\left(s\right){g}^{\prime }\left(s\right)|\le {2}^{\frac{1-p}{p}}$

1) ${\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)$$\left(0,\infty \right)$ 递减；

2) $\underset{s\to {0}^{+}}{\mathrm{lim}}{\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)=+\infty$

3) 存在常数 ${C}_{1}>0$ 使得对所有 $s\ge 1$，成立 $0<{\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)\le {C}_{1}$

3. 定理1.1的证明

1) 对几乎处处 $x\in \Omega$，有 $0<\underset{_}{v}\left(x\right)\le \stackrel{¯}{v}\left(x\right)$

2) 对任意 $\phi \in {W}_{0}^{1,p}\left(\Omega \right)$$\phi \ge 0$，有

${\int }_{\Omega }{|\nabla \underset{_}{v}|}^{p-2}\nabla \underset{_}{v}\nabla \phi \le {\int }_{\Omega }b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\phi +{\int }_{\Omega }f\left(x,g\left(\underset{_}{v}\right)\right){g}^{\prime }\left(\underset{_}{v}\right)\phi ,$

${\int }_{\Omega }{|\nabla \stackrel{¯}{v}|}^{p-2}\nabla \stackrel{¯}{v}\nabla \phi \ge {\int }_{\Omega }b\left(x\right){\left(g\left(\stackrel{¯}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\stackrel{¯}{v}\right)\phi +{\int }_{\Omega }f\left(x,g\left(\stackrel{¯}{v}\right)\right){g}^{\prime }\left(\stackrel{¯}{v}\right)\phi .$

$0<\underset{_}{v}\left(x\right)\le v\left(x\right)\le \stackrel{¯}{v}\left(x\right).$

$\left\{\begin{array}{l}-{\Delta }_{p}u=\psi \left(x\right)\text{}x\in \Omega ,\\ u=0\text{}x\in \partial \Omega .\end{array}$ (3.1)

$u>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \Omega ;\text{}\frac{\partial u}{\partial \nu }>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in \partial \Omega \text{,}$

1) $b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\in {L}^{q}\left(\Omega \right)$${‖\underset{_}{v}‖}_{\infty }\le \delta /2$，其中 $\delta >0$ 且由(f1)给定；

2) 对几乎处处 $x\in \Omega$，有 $0<\underset{_}{v}\left(x\right)\le \stackrel{¯}{v}\left(x\right)$

3) $\underset{_}{v},\stackrel{¯}{v}$ 分别为方程(1.5)上解和下解。

$\left\{\begin{array}{l}-{\Delta }_{p}\omega =b\left(x\right),\text{}x\in \Omega ,\\ \omega =0,\text{}x\in \partial \Omega ,\end{array}$ (3.2)

${\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)\ge k+1,$ (3.3)

$b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\in {L}^{q}\left(\Omega \right)$${‖\underset{_}{v}‖}_{\infty }\le \delta /2.$ (3.4)

$\left\{\begin{array}{l}-{\Delta }_{p}z=b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+2{F}_{0}\text{}x\in \Omega ,\\ z=0\text{}x\in \partial \Omega ,\end{array}$ (3.5)

${\left(g\left(\underset{_}{v}\left(x\right)\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\left(x\right)\right)\ge k+1.$ (3.6)

$\stackrel{¯}{v}=z$，对所有 $\phi \in {W}_{0}^{1,p}\left(\Omega \right)$$\phi \ge 0$，可知

$\begin{array}{c}{\int }_{\Omega }{|\nabla \stackrel{¯}{v}|}^{p-2}\nabla \stackrel{¯}{v}\nabla \phi \ge {\int }_{\Omega }b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\phi \ge {\int }_{\Omega }b\left(x\right)\phi \\ \ge {\epsilon }^{p-1}{\int }_{\Omega }b\left(x\right)\phi ={\int }_{\Omega }{|\nabla \underset{_}{v}|}^{p-2}\nabla \underset{_}{v}\nabla \phi .\end{array}$ (3.7)

$\begin{array}{l}{\int }_{\Omega }{|\nabla \underset{_}{v}|}^{p-2}\nabla \underset{_}{v}\nabla \phi -{\int }_{\Omega }b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\phi -{\int }_{\Omega }f\left(x,g\left(\underset{_}{v}\right)\right){g}^{\prime }\left(\underset{_}{v}\right)\phi \\ \le {\int }_{\Omega }b\left(x\right)\left({\epsilon }^{p-1}+k-{\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\right)\phi \le 0,\end{array}$

${‖\stackrel{¯}{v}‖}_{\infty }\le {C}_{0}.$ (3.8)

$\begin{array}{l}{\int }_{\Omega }{|\nabla \stackrel{¯}{v}|}^{p-2}\nabla \stackrel{¯}{v}\nabla \phi -{\int }_{\Omega }b\left(x\right){\left(g\left(\stackrel{¯}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\stackrel{¯}{v}\right)\phi -{\int }_{\Omega }f\left(x,g\left(\stackrel{¯}{v}\right)\right){g}^{\prime }\left(\stackrel{¯}{v}\right)\phi \\ \ge {\int }_{\Omega }b\left(x\right)\left({\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{\stackrel{¯}{v}}\right)-{\left(g\left(\stackrel{¯}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\stackrel{¯}{v}\right)\right)\phi +{\int }_{\Omega }\left(2{F}_{0}-f\left(x,g\left(\stackrel{¯}{v}\right)\right)\right){g}^{\prime }\left(\stackrel{¯}{v}\right)\phi \\ \ge {\int }_{\Omega }\left(2{F}_{0}-f\left(x,g\left(\stackrel{¯}{v}\right)\right)\right){g}^{\prime }\left(\stackrel{¯}{v}\right)\phi \\ \ge {F}_{0}{\int }_{\Omega }\left(1-{a}_{1}\left(x\right){\left(g\left(\stackrel{¯}{v}\right)\right)}^{r-1}\right){g}^{\prime }\left(\stackrel{¯}{v}\right)\phi .\end{array}$ (3.9)

${\alpha }_{0}={C}_{0}^{1-r}>0$ ( ${C}_{0}$ 为(3.8)式中的常数)，若 ${‖{a}_{1}‖}_{\infty }\le {\alpha }_{0}$，可知对几乎处处 $x\in \Omega$，有

${a}_{1}\left(x\right)\le \frac{1}{{C}_{0}^{r-1}}\le \frac{1}{{‖\stackrel{¯}{v}‖}_{\infty }^{r-1}}\le \frac{1}{{\left(g\left({‖\stackrel{¯}{v}‖}_{\infty }\right)\right)}^{r-1}}\le \frac{1}{{\left(g\left(\stackrel{¯}{v}\left(x\right)\right)\right)}^{r-1}},$

$\stackrel{˜}{h}\left(x,s\right)=\left\{\begin{array}{l}b\left(x\right){\left(g\left(\underset{_}{v}\left(x\right)\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\left(x\right)\right)+f\left(x,g\left(\underset{_}{v}\left(x\right)\right)\right){g}^{\prime }\left(\underset{_}{v}\left(x\right)\right),\text{}s<\underset{_}{v}\left(x\right),\\ b\left(x\right){\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)+f\left(x,g\left(s\right)\right){g}^{\prime }\left(s\right),\text{}\underset{_}{v}\left(x\right)\le s\le \stackrel{¯}{v}\left(x\right),\\ b\left(x\right){\left(g\left(\stackrel{¯}{v}\left(x\right)\right)\right)}^{-\gamma }{g}^{\prime }\left(\stackrel{¯}{v}\left(x\right)\right)+f\left(x,g\left(\stackrel{¯}{v}\left(x\right)\right)\right){g}^{\prime }\left(\stackrel{¯}{v}\left(x\right)\right),\text{}s>\stackrel{¯}{v}\left(x\right),\end{array}$ (3.10)

$\left\{\begin{array}{l}-{\Delta }_{p}v=\stackrel{˜}{h}\left(x,v\right)\text{}x\in \Omega ,\\ v=0\text{}x\in \partial \Omega ,\end{array}$ (3.11)

$\stackrel{˜}{\Phi }\left(v\right)=\frac{1}{p}{\int }_{\Omega }{|\nabla v|}^{p}-{\int }_{\Omega }\stackrel{˜}{H}\left(x,v\right),$ (3.12)

$|\stackrel{˜}{h}\left(x,s\right)|\le b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{M}_{1},$ (3.13)

$\stackrel{˜}{\Phi }\left(v\right)\ge \frac{1}{p}{‖v‖}^{p}-{C}_{1}{‖v‖}_{\frac{q}{q-1}}-{C}_{2}{‖v‖}_{1}\ge \frac{1}{p}{‖v‖}^{p}-{C}_{3}‖v‖.$

$\left\{\begin{array}{l}{v}_{n}\stackrel{w}{\to }v\text{}\left(在{W}_{0}^{1,p}\left(\Omega \right)中\right),\\ {v}_{n}\underset{}{\to }v\text{}\left(在{L}^{\sigma }\left(\Omega \right)中,\text{}1\le \sigma <{p}^{*}\right),\\ {v}_{n}\stackrel{}{\to }v\text{}a.e.\text{}x\in \Omega .\end{array}$ (3.14)

${\int }_{\Omega }\stackrel{˜}{H}\left(x,{v}_{n}\right)\to {\int }_{\Omega }\stackrel{˜}{H}\left(x,v\right).$

4. 定理1.2的证明

$〈A\left(u\right),\eta 〉={\int }_{\Omega }{|\nabla u|}^{p-2}\nabla u\nabla \eta ,$

$\stackrel{^}{h}\left(x,s\right)=\left\{\begin{array}{l}b\left(x\right){\left(g\left(\underset{_}{v}\left(x\right)\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\left(x\right)\right)+f\left(x,g\left(\underset{_}{v}\left(x\right)\right)\right){g}^{\prime }\left(\underset{_}{v}\left(x\right)\right),\text{}s<\underset{_}{v}\left(x\right),\\ b\left(x\right){\left(g\left(s\right)\right)}^{-\gamma }{g}^{\prime }\left(s\right)+f\left(x,g\left(s\right)\right){g}^{\prime }\left(s\right),\text{}s\ge \underset{_}{v}\left(x\right),\end{array}$ (4.1)

$\left\{\begin{array}{l}-{\Delta }_{p}v=\stackrel{^}{h}\left(x,v\right)\text{}x\in \Omega ,\\ v=0\text{}x\in \partial \Omega ,\end{array}$ (4.2)

$\stackrel{^}{\Phi }\left(v\right)=\frac{1}{p}{\int }_{\Omega }{|\nabla v|}^{p}-{\int }_{\Omega }\stackrel{^}{H}\left(x,v\right),$ (4.3)

$|\stackrel{^}{h}\left(x,s\right)|\le b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{M}_{1}.$

$|\stackrel{^}{h}\left(x,s\right)|\le b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{C}^{*}\left(1+{a}_{1}\left(x\right){|s|}^{\frac{r-2}{2}}\right).$

$|\stackrel{^}{h}\left(x,s\right)|\le b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{C}^{*\text{*}}\left(1+{a}_{1}\left(x\right){|s|}^{\frac{r-2}{2}}\right),$

$|\stackrel{^}{H}\left(x,s\right)|\le {l}_{1}\left(x\right)s+{C}^{**}{a}_{1}\left(x\right){|s|}^{\frac{r}{2}},$ (4.4)

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge \left(1-\theta \right)s\left(b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{M}_{1}\right).$ (4.5)

${m}_{0}=\underset{\left(x,s\right)\in \Omega ×\left(0,{s}_{0}\right)}{\text{esssup}}|f\left(x,s\right)|$${M}_{0}=\underset{\left(x,s\right)\in \Omega ×\left(0,{s}_{0}\right)}{\text{esssup}}|F\left(x,s\right)|$。不失一般性，根据条件(f3)我们假设 ${s}_{0}>g\left({‖\underset{_}{v}‖}_{\infty }\right)$，则当 $\underset{_}{v}\left(x\right)\le s<{g}^{-1}\left({s}_{0}\right)$ 时，由引理2.2-(6)和引理2.3-(1)，可以推出

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge -{m}_{0}{s}_{0}-\theta {m}_{0}s-\frac{3-\gamma }{1-\gamma }\theta b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)s-2\theta {M}_{0}.$ (4.6)

$s\ge {g}^{-1}\left({s}_{0}\right)$ 时，根据条件(f3)，进一步得到

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge -\theta {m}_{0}s-\frac{3-\gamma }{1-\gamma }\theta b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)s-\theta {M}_{0}.$ (4.7)

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge -{C}_{1}-{C}_{2}{\omega }_{1}\left(x\right)s,$

$C+{o}_{n}\left(1\right)‖{v}_{n}‖\ge \theta \stackrel{^}{\Phi }\left({v}_{n}\right)-〈{\stackrel{^}{\Phi }}^{\prime }\left({v}_{n}\right),{v}_{n}〉=\left(\frac{\theta }{p}-1\right){‖{v}_{n}‖}^{p}+{\int }_{\Omega }\left(\stackrel{^}{h}\left(x,{v}_{n}\right){v}_{n}-\theta \stackrel{^}{H}\left(x,{v}_{n}\right)\right),$

$C+{C}_{1}|\Omega |+{C}_{2}{‖{\omega }_{1}‖}_{q}‖{v}_{n}‖+{o}_{n}\left(1\right)‖{v}_{n}‖\ge \left(\frac{\theta }{p}-1\right){‖{v}_{n}‖}^{p}.$

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge -2\theta b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-1}{g}^{\prime }\left(\underset{_}{v}\right)s-\left(1+\theta \right){m}_{0}s-2\theta {M}_{0}.$ (4.8)

$s\ge {g}^{-1}\left({s}_{0}\right)$，利用(f3)即得

$\stackrel{^}{h}\left(x,s\right)s-\theta \stackrel{^}{H}\left(x,s\right)\ge -2\theta b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-1}{g}^{\prime }\left(\underset{_}{v}\right)s-\theta {m}_{0}s-\theta {M}_{0}.$ (4.9)

$C+{C}_{1}|\Omega |+{C}_{2}{‖{\omega }_{1}‖}_{q}‖{v}_{n}‖+{o}_{n}\left(1\right)‖{v}_{n}‖\ge \left(\frac{\theta }{p}-1\right){‖{v}_{n}‖}^{p}.$

$\left\{\begin{array}{l}{v}_{n}\stackrel{w}{\to }v\text{}\left(在{W}_{0}^{1,p}\left(\Omega \right)中\right),\\ {v}_{n}\underset{}{\to }v\text{}\left(在{L}^{\sigma }\left(\Omega \right)中,\text{}1\le \sigma <{p}^{*}\right),\\ {v}_{n}\stackrel{}{\to }v\text{}a.e.\text{}x\in \Omega .\end{array}$ (4.10)

${o}_{n}\left(1\right)=〈{\stackrel{^}{\Phi }}^{\prime }\left({v}_{n}\right),\eta 〉={\int }_{\Omega }{|\nabla {v}_{n}|}^{p-2}\nabla {v}_{n}\nabla \eta -{\int }_{\Omega }\stackrel{^}{h}\left(x,{v}_{n}\right)\eta .$

${o}_{n}\left(1\right)=〈{\stackrel{^}{\Phi }}^{\prime }\left({v}_{n}\right),{v}_{n}-v〉={\int }_{\Omega }{|\nabla {v}_{n}|}^{p-2}\nabla {v}_{n}\nabla \left({v}_{n}-v\right)-{\int }_{\Omega }\stackrel{^}{h}\left(x,{v}_{n}\right)\left({v}_{n}-v\right).$ (4.11)

$\begin{array}{c}|{\int }_{\Omega }\stackrel{^}{h}\left(x,{v}_{n}\right)\left({v}_{n}-v\right)|\le {\int }_{\Omega }|b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)+{C}_{1}\left(1+{a}_{1}\left(x\right)\right){|{v}_{n}|}^{\frac{r-2}{2}}||{v}_{n}-v|\\ \le {‖b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)‖}_{q}{‖{v}_{n}-v‖}_{\frac{q}{q-1}}+C{‖{v}_{n}‖}_{r-1}^{\frac{r-1}{2}}{‖{v}_{n}-v‖}_{\frac{2\left(r-1\right)}{r}}+C{‖{v}_{n}-v‖}_{1}.\end{array}$

$\underset{n\to \infty }{\mathrm{lim}}{\int }_{\Omega }\stackrel{^}{h}\left(x,{v}_{n}\right)\left({v}_{n}-v\right)=0.$ (4.12)

$\underset{n\to \infty }{\mathrm{lim}}〈A\left({v}_{n}\right),{v}_{n}-v〉=0.$

1) 存在 ${R}_{1}>0,\text{\hspace{0.17em}}\rho >0$ 使得 $\underset{v\in \partial {B}_{{R}_{1}}\left(0\right)}{\mathrm{inf}}\stackrel{^}{\Phi }\left(v\right)\ge \rho$

2) 存在 $e\in {W}_{0}^{1,p}\left(\Omega \right)\{B}_{{R}_{1}}\left(0\right)$ 使得 $\stackrel{^}{\Phi }\left(e\right)<0$

${\int }_{\Omega }\stackrel{^}{H}\left(x,v\right)\le {‖{l}_{1}‖}_{q}{‖v‖}_{\frac{q}{q-1}}+C{‖{a}_{1}‖}_{\infty }{‖v‖}_{\frac{r}{2}}^{\frac{r}{2}}.$

$\stackrel{^}{\Phi }\left(v\right)\ge \frac{1}{p}{‖v‖}^{p}-{C}_{1}{‖{l}_{1}‖}_{q}‖v‖+{C}_{2}{‖{a}_{1}‖}_{\infty }{‖v‖}^{\frac{r}{2}}.$

$F\left(x,s\right)\ge \frac{F\left(x,{s}_{0}\right)}{{s}_{0}^{2\theta }}{s}^{2\theta }-Q\left(x\right).$ (4.13)

$\stackrel{^}{H}\left(x,s\underset{_}{v}\right)\ge \stackrel{^}{H}\left(x,\underset{_}{v}\right)+F\left(x,g\left(s\underset{_}{v}\right)\right)-F\left(x,g\left(v\right)\right).$

$\stackrel{^}{\Phi }\left(s\underset{_}{v}\right)\le \frac{{s}^{p}}{p}{‖\underset{_}{v}‖}^{p}-\frac{{s}^{\theta }}{{s}_{0}^{2\theta }}{\int }_{\Omega }F\left(x,{s}_{0}\right){\left(g\left(\underset{_}{v}\right)\right)}^{2\theta }-{C}^{*}.$ (4.14)

${\int }_{\Omega }{|\nabla \stackrel{^}{v}|}^{p-2}\nabla \stackrel{^}{v}\nabla {\left(\underset{_}{v}-\stackrel{^}{v}\right)}^{+}={\int }_{\left\{\stackrel{^}{v}\le \underset{_}{v}\right\}}\stackrel{^}{h}\left(x,\stackrel{^}{v}\right){\left(\underset{_}{v}-\stackrel{^}{v}\right)}^{+}\ge {\int }_{\Omega }{|\nabla \underset{_}{v}|}^{p-2}\nabla \underset{_}{v}\nabla {\left(\underset{_}{v}-\stackrel{^}{v}\right)}^{+}.$

$\stackrel{^}{\Phi }\left(\stackrel{˜}{v}\right)=\stackrel{˜}{\Phi }\left(\stackrel{˜}{v}\right)\le \stackrel{˜}{\Phi }\left(\underset{_}{v}\right)=\stackrel{^}{\Phi }\left(\underset{_}{v}\right),$ (4.15)

$\stackrel{^}{\Phi }\left(\underset{_}{v}\right)=\frac{1}{p}{‖\underset{_}{v}‖}^{p}-{\int }_{\Omega }b\left(x\right){\left(g\left(\underset{_}{v}\right)\right)}^{-\gamma }{g}^{\prime }\left(\underset{_}{v}\right)\underset{_}{v}+f\left(x,g\left(\underset{_}{v}\right)\right){g}^{\prime }\left(\underset{_}{v}\right)\underset{_}{v}\le \left(\frac{1}{p}-1\right){‖\underset{_}{v}‖}^{p}<0.$ (4.16)

$\stackrel{˜}{v}$ 替换(4.1)式中的 $\underset{_}{v}$，运用与上述证明相同的方法，我们可进一步推出，对几乎处处 $x\in \Omega$，均成立 $\stackrel{˜}{v}\le \stackrel{^}{v}$

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