伊犁师范大学数学与统计学院，新疆 伊宁

收稿日期：2021年9月21日；录用日期：2021年10月14日；发布日期：2021年10月22日

摘要

拟线性椭圆方程问题是数学学科中重要的研究内容之一。椭圆型微分方程解的存在性问题近年来得到人们的广泛关注，基于文献研究，本文研究一类拟线性椭圆方程边值问题多解的存在性，并验证了一系列引理和定理，运用Nehari流形和纤维映射方法证明了该问题至少有两个正解。

关键词

Nehari流形，纤维映射，椭圆方程

Existence of Multiple Solutions for Boundary Value Problems of a Class of Quasilinear Elliptic Equations

Yimei Tian^*, Lin Chen^#

School of Mathematics and Statistic, Yili Normal University, Yining Xingjiang

Received: Sep. 21^st, 2021; accepted: Oct. 14^th, 2021; published: Oct. 22^nd, 2021

ABSTRACT

The problem of quasilinear elliptic equation is one of the important research contents in mathematics. The existence of solutions for elliptic differential equations has attracted extensive attention in recent years. Based on literature research, this paper studies the existence of multiple solutions for a class of quasilinear elliptic equation boundary value problems, verifies a series of lemmas and theorems, and proves that the problem has at least two positive solutions by using Nehari manifold and fiber mapping method.

Keywords:Nehari Manifold, Fiber Mapping, Elliptic Equations

Copyright © 2021 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

近年来，椭圆型偏微分方程解的存在性问题得到人们的广泛的关注 [1] [2] [3]。Chen在文献 [4] 中运用Nehari流形和纤维映射的方法，研究了一类非线性边值问题

$\{\begin{array}{l}-div\left({\left|x\right|}^{-ap}{\left|\nabla u\right|}^{p-2}\nabla u\right)=\lambda h\left(x\right){\left|u\right|}^{m-2}u+H\left(x\right){\left|u\right|}^{n-2}u,x\in \Omega ,\\ u\left(x\right)=0,x\in \partial \Omega ,\end{array}$

多解的存在性，其中 $\Omega \subset R^N\left(N\ge 3\right)$ 是具有光滑边界的有界区域， $\lambda >0$，$0\in \Omega$，$1<p<N$，$0\le a<\frac{N-p}{p}$，$1<m<p<n<\frac{pN}{N-\left(1+a\right)p}$，$h\left(x\right)$ 和 $g\left(x\right)$ 是在 $\Omega$ 上符号会发生改变的Lebesgue可测函数。Brown和Wu在文献 [5] 中运用Nehari流形和纤维映射方法证明了一类半线性椭圆边值问题

$\{\begin{array}{l}-\Delta u\left(x\right)=\lambda h\left(x\right)u^q+H\left(x\right)u^p,x\in \Omega ,\\ u\left(x\right)=0,x\in \partial \Omega ,\end{array}$

至少有两个正解，其中 $\Omega \subset R^N$ 是一个具有光滑边界的有界区域， $0<q<1<p<\frac{N+2}{N-2}$，$\lambda >0$, 函数 $h\left(x\right)$ 和 $H\left(x\right)$ 在 $\Omega$ 中可以改变符号。

受文献 [4] [5] 的启发，本文研究一类拟线性椭圆边值问题

$\{\begin{array}{l}-{\Delta }_{p}u\left(x\right)=h\left(x\right){\left|u\right|}^{m-2}u+\lambda g\left(x\right){\left|u\right|}^{r-2}u,x\in \Omega ,\\ u\left(x\right)=0,x\in \partial \Omega ,\end{array}$ (1)

正解的存在性，其中 $\lambda >0$，参数 $p,m,r$ 满足 $1<r<p<m<p^{*}$，$p^{*}=\frac{pN}{N-p}$，这里 $p^{*}$ 是Sobolev临界指数 [6]。本文运用Nehari流形和变分方法研究问题(1)多解的存在性。

为研究问题的方便，假设函数 $h\left(x\right)$ 和 $g\left(x\right)$ 满足以下条件：

(A₁) $h\left(x\right)\in L^{\alpha }\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$ 且 $\alpha =\frac{p^{*}}{p^{*}-m}$。

(A₂) $g\left(x\right)\in L^{\beta }\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$ 且 $\beta =\frac{p^{*}}{p^{*}-r}$。

设 $X\equiv W_0^{1,p}\left(\Omega \right)$ 是空间 $C_0^{\infty }\left(\Omega \right)$ 关于范数

$\Vert u\Vert ={\left({\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}\right)}^{\frac{1}{p}}$

的完备化空间。

2. 基本定理

定义1设 $u\in X$，若对任意 $\phi \in C_0^{\infty }\left(\Omega \right)$，有

${\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^{p-2}\nabla u\nabla \phi \text{d}x}-{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^{m-2}u\phi \text{d}x}-\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^{r-2}u\phi \text{d}x}=0$.

成立，则称u是问题(1)的弱解。

问题(1)所对应的能量泛函为

$J_{\lambda }\left(u\right)=\frac{1}{p}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\frac{1}{m}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-\frac{1}{r}{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}$.

显然，问题(1)的解对应能量泛函 $J_{\lambda }\left(u\right)$ 的临界点。

由于泛函 $J_{\lambda }\left(u\right)$ 在空间X上无界，从而引入Nehari流形

$M_{\lambda }=\left\{u\in X\backslash \left\{0\right\}:\langle {J^{\prime }}_{\lambda }\left(u\right),u\rangle \right\}=0$,

其中 $\langle ,\rangle$ 代表普通对偶，则 $u\in M_{\lambda }$ 当且仅当

${\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}={\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}+\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^{r-2}u\phi \text{d}x}$

成立。从而，当 $u\in M_{\lambda }$ 时，有

$J_{\lambda }\left(u\right)=\left(\frac{1}{p}-\frac{1}{r}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(\frac{1}{m}-\frac{1}{r}\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$ (2)

$=\left(\frac{1}{p}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(\frac{1}{r}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}$. (3)

构造纤维映射 ${\varnothing }_u:t\in \Omega \to J_u\left(tu\right)$，$\forall u\in X$，易见 $u\in M_{\lambda }$ 当且仅当 ${{\varnothing }^{\prime }}_u\left(1\right)=0$。将集合 $M_{\lambda }$ 分成三部分，分别定义为

$M_{\lambda }^{+}=\left\{u\in M_{\lambda }|{{\varnothing }^{″}}_u\left(1\right)>0\right\}$ ;

$M_{\lambda }^{-}=\left\{u\in M_{\lambda }|{{\varnothing }^{″}}_u\left(1\right)<0\right\}$ ;

$M_{\lambda }^0=\left\{u\in M_{\lambda }|{{\varnothing }^{″}}_u\left(1\right)=0\right\}$.

引理1假定(A₁)-(A₂)成立，函数 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }$ 上强制且有界。

证明：由于 $g\left(x\right)\in L^{\beta }\left(\Omega \right)\cap L^{\infty }\left(\Omega \right)$，$\beta =\frac{p^{*}}{p^{*}-r}$，由Hölder不等式可得到

$\begin{array}{c}{\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}\le {\left({\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{\beta }\text{d}x}\right)}^{\frac{1}{\beta }}{\left({\displaystyle {\int }_{\Omega }{\left|u\right|}^p\text{d}x}\right)}^{\frac{r}{p}}\\ \le c_1{\left({\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{\beta }\text{d}x}\right)}^{\frac{1}{\beta }}{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^{p^{*}}\text{d}x}\right)}^{\frac{r}{p^{*}}}\\ \le c_{\beta }{\Vert u\Vert }^n\end{array}$, (4)

其中 $c_{\beta }={\left({\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{\beta }\text{d}x}\right)}^{\frac{1}{\beta }}$。

类似地，有

${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le c_{\alpha }{\left|\left|u\right|\right|}^m$, (5)

其中 $c_{\alpha }={\left({\displaystyle {\int }_{\Omega }{\left|g\left(x\right)\right|}^{\alpha }\text{d}x}\right)}^{\frac{1}{\alpha }}$，$\alpha =\frac{p^{*}}{p^{*}-m}$。则由(2)~(5)可得

$\begin{array}{l}{J}_{\lambda }\left(u\right)=\left(\frac{1}{p}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(\frac{1}{m}-\frac{1}{r}\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\\ \ge J_{\lambda }\left(u\right)=\left(\frac{1}{p}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(\frac{1}{m}-\frac{1}{r}\right)c_{\alpha }{\Vert u\Vert }^m.\end{array}$

当 $\Vert u\Vert \to \infty$ 时，有 $J_{\lambda }\left(u\right)\to \infty$。则函数 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }$ 上强制且有界。

引理2假定(A₁)~(A₂)成立。存在 ${\lambda }_0>0$ 使得对任意 $\lambda \in \left(0,{\lambda }_0\right)$ 有 $M_{\lambda }^0=\varnothing$。

证明：令 ${\lambda }_0={\left(\frac{r-p}{\left(r-m\right)c_{\alpha }}\right)}^{\frac{p-r}{m-p}}\frac{p-m}{\left(r-m\right)c_{\beta }}$，假设结论不成立，则存在 $\lambda \in \left(0,{\lambda }_0\right)$ 使得 $M_{\lambda }^0\ne \varnothing$，从而 $\exists u\in M_{\lambda }^0$ 有

$\begin{array}{c}0={{\varnothing }^{″}}_u\left(1\right)=\left(p-r\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(r-m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\\ =\left(p-m\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(m-r\right){\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}.\end{array}$ (6)

将(5)代入(6)中可得到，

$\left(r-p\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}=\left(r-m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le \left(r-m\right)c_{\alpha }{\Vert u\Vert }^m$,

$\left(p-m\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}=\left(r-m\right){\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}\le \left(r-m\right)c_{\beta }{\Vert u\Vert }^n$.

经计算可得

${\left(\frac{r-p}{\left(r-m\right)c_{\alpha }}\right)}^{\frac{1}{m-p}}\le \Vert u\Vert \le {\left(\frac{\lambda \left(r-m\right)}{p-m}{c}_{\beta }\right)}^{\frac{1}{p-r}}$,

从而 $\lambda \ge {\lambda }_0$，矛盾！因此，存在 ${\lambda }_0>0$ 使得对任意 $\lambda \in \left(0,{\lambda }_0\right)$ 有 $M_{\lambda }^0=\varnothing$。证毕。

引理3假定(A₁)~(A₂)成立。若 $u_0$ 是 $J_{\lambda }\left(u\right)$ 在 $M_{\lambda }\left(\Omega \right)$ 上的局部极小值或局部极大值且 $u_0\notin M_{\lambda }^0$，则 $u_0$ 是泛函 $J_{\lambda }\left(u\right)$ 上的一个临界点。

证明：令

$S\left(u\right)={\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-{\displaystyle {\int }_{\Omega }\lambda g}\left(x\right){\left|u\right|}^r\text{d}x,u\in X$.

考虑最优化问题

$\min \left\{J_{\lambda }\left(u\right)|u\in M_{\lambda }\left(\Omega \right),S\left(u\right)=0\right\}$.

由Lagrange乘子理论，存在 $\mu \in ℝ$ 使得 ${J^{\prime }}_{\lambda }\left(u_0\right)=\mu S^{\prime }\left(u_0\right)$，从而

$\langle {J^{\prime }}_{\lambda }\left(u_0\right),u_0\rangle =\mu \langle S^{\prime }\left(u_0\right),u_0\rangle$. (7)

由于 $u_0\in M_{\lambda }$，则

$\langle {J^{\prime }}_{\lambda }\left(u_0\right),u_0\rangle =0$.

此外

$\begin{array}{c}\langle S^{\prime }\left(u_0\right),u_0\rangle =p{\Vert u_0\Vert }^p-m{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u_0\right|}^m\text{d}x}-r{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u_0\right|}^r\text{d}x}\\ =\left(p-r\right){\left|\left|u_0\right|\right|}^p-\left(m-r\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u_0\right|}^m\text{d}x}\\ =\left(p-m\right){\left|\left|u_0\right|\right|}^p-\left(m+r\right){\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u_0\right|}^r\text{d}x}\end{array}$.

因此， $u_0\notin M_{\lambda }^0$，$\langle S^{\prime }\left(u_0\right),u_0\rangle \ne 0$。由(7)可得到 $\mu =0$,${J^{\prime }}_{\lambda }\left(u_0\right)=0$。从而， $u_0$ 是 $J_{\lambda }\left(u\right)$ 的一个临界点。证毕。

接下来探究纤维映射 ${\varnothing }_u\left(t\right)$ 的性质。考虑函数

$F_u\left(t\right)=t^{p-r}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-t^{m-r}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$.

显然，对 $t>0$，$tu\in M_{\lambda }\left(\Omega \right)$ 当且仅当t是下面方程的解

$F_u\left(t\right)={\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}$. (8)

此外，若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$，则有

${F^{\prime }}_u\left(t\right)=\left(p-r\right)t^{p-r-1}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(m-r\right)t^{m-r-1}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\ge 0$. (9)

其中 $1<r<p<m$。因此，当 $t\ge 0$ 时 $F_u\left(t\right)$ 是严格递增的。

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$，由 ${F^{\prime }}_u\left(t\right)=0$ 可以得到唯一临界点

$t_0={\left(\frac{\left(p-r\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}}{\left(m-r\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}}\right)}^{\frac{1}{m-p}}$.

由于 $1<r<p<m$，当 $t\to 0^{+}$ 时，有 $M_u\left(t\right)\to 0$，当 $t\to +\infty$ 时，有 $M_u\left(t\right)\to -\infty$。经计算可得

${F^{″}}_u\left(t\right)=\left(p-r\right)\left(p-r-1\right)t^{p-r-2}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\left(m-r\right)\left(m-r-1\right)t^{m-r-2}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$,

${F^{″}}_u\left(t_0\right)=\left(m-r\right)\left(p-m\right)t_0^{m-r-2}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}<0$.

这表明函数 $M_u\left(t\right)$ 在 $\left(0,t_0\right)$ 上单调递增，在 $\left(t_0,+\infty \right)$ 上单调递减。若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$，则 $t_0$ 是唯一的极大值点。

假定 $tu\in M_{\lambda }\left(\Omega \right)$，由(9)可得 ${{\varnothing }^{″}}_{tu}\left(1\right)=t^{r+1}{F^{\prime }}_u\left(t\right)$。若 ${F^{\prime }}_u\left(t\right)>0$，则 $tu\in M_{\lambda }^{+}\left(\Omega \right)$。若 ${F^{\prime }}_u\left(t\right)<0$，则 $tu\in M_{\lambda }^{-}\left(\Omega \right)$。

接下来，通过探究 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$ 和 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}$ 的符号，研究纤维映射 ${\varnothing }_u\left(t\right)$ 的特征。

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$ 且 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}\le 0$，则 ${\varnothing }_u\left(t\right)$ 是关于t的递增函数。

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$ 且 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}>0$，则存在 $t=t\left(u\right)$ 使得 ${{\varnothing }^{\prime }}_u\left(t\right)=0$，$t\left(u\right)u\in M_{\lambda }\left(\Omega \right)$，从而有

${F^{\prime }}_u\left(t\right)=\lambda \left(p-r\right)t^{-1}{\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}+\left(p-m\right)t^{m-r-1}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$.

因此 $t\left(u\right)u\in M_{\lambda }^{+}\left(\Omega \right)$。从而，由于 ${\lim }_{t\to +\infty }{\varnothing }_u\left(t\right)=+\infty$，从而 ${\varnothing }_u\left(t\right)$ 先递减再递增。因此 ${\varnothing }_u\left(t\right)$ 在 $t=t\left(u\right)$ 处有唯一临界点且为局部极小值点。

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$ 且 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}\le 0$，则函数 $F_u\left(t\right)$ 先递增再递减，有一个极大值点。则方程(8)有一个正解，从而有唯一的 $t\left(u\right)>0$ 使得 $t\left(u\right)u\in M_{\lambda }\left(\Omega \right)$ 且 ${M^{\prime }}_u\left(t\right)<0$。因此 $t\left(u\right)u\in S_{\lambda }^{-}\left(\Omega \right)$。由于 ${\lim }_{t\to +\infty }{\varnothing }_u\left(t\right)=-\infty$，则 ${\varnothing }_u\left(t\right)$ 先递增再递减，由此可得纤维映射 ${\varnothing }_u\left(t\right)$ 有唯一的一个临界点且为局部极大值点。

最后考虑情形 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$，${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}>0$。若 $\lambda >0$ 足够大，则(8)无解，因此 ${\varnothing }_u\left(t\right)$ 无临界点，在此情形 ${\varnothing }_u\left(t\right)$ 是一个递减函数。若 $\lambda >0$ 充分小，则在(8)有两个解 $t_1\left(u\right)<t_2\left(u\right)$ 且 ${M^{\prime }}_u\left(t_1\right)>0$，${M^{\prime }}_u\left(t_2\right)>0$。因此 $t_1\left(u\right)u\in M_{\lambda }^{+}\left(t\right)$，$t_2\left(u\right)u\in M_{\lambda }^{-}\left(t\right)$。从而 ${\varnothing }_u\left(t\right)$ 在 $\left(0,t_0\right)$ 上单调递减，在 $\left(t_1,t_2\right)$ 上单调递增，在 $\left(t_2,+\infty \right)$ 上单调递减。

引理4假定(A₁)~(A₂)成立。存在 ${\lambda }_1>0,t_0>0$，使得对任意 $u\in X\backslash \left\{0\right\}$ 和 $0<\lambda <{\lambda }_1$ 有 ${\varnothing }_u\left(t_0\right)>0$。

证明：若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$，则当t充分大时 ${\varnothing }_u\left(t_0\right)>0$。若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$，记

${\phi }_u\left(t\right)=\frac{t^p}{p}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\frac{t^m}{m}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$,

则 ${\phi }_u\left(0\right)=0$。当 $t\to +\infty$ 时， ${\phi }_u\left(t\right)=-\infty$。此外

${{\phi }^{\prime }}_u\left(t\right)=t^{p-1}{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-t^{m-1}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$.

经计算可得， ${\phi }_u\left(t\right)$ 在 $t_1$ 处有最大值 ${\phi }_u\left(t_1\right)=\left(\frac{1}{p}-\frac{1}{m}\right)t_1^{\frac{1}{m-p}}$，其中 $t_1=\frac{{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}}{{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}}$。

由引理1可知

$\begin{array}{c}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}={\displaystyle {\int }_{\Omega }h\left(x\right){\left|x\right|}^r{\left|x\right|}^{-r}{\left|u\right|}^m\text{d}x}\\ \le h_r{\displaystyle {\int }_{\Omega }{\left|x\right|}^{-r}{\left|u\right|}^m\text{d}x}\\ \le h_r{c}_{r,m}^{-\frac{m}{p}}{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}\right)}^{\frac{m}{p}},\end{array}$

其中 $r\le m+N\left(1-\frac{m}{p}\right)$，$h_r={\Vert h\left(x\right)\Vert }_{\infty }$ 且 $c_{r,m}$ 代表嵌入 $X\to L^m\left(\Omega \right)$ 的Sobolve常数。则

${\phi }_u\left(t_1\right)\ge \left(\frac{1}{p}-\frac{1}{m}\right){\left(h_r{c}_{r,m}^{-\frac{m}{p}}\right)}^{\frac{p}{p-m}}\equiv \delta >0$,

其中 $\delta$ 独立于u。同理有

$\begin{array}{c}\frac{t_1^r}{r}{\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}\le \frac{t_1^r}{r}{\Vert g\Vert }_{\infty }{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}\right)}^{\frac{r}{p}}{c}_{\alpha ,r}^{-\frac{r}{p}}\\ =\frac{g_{\alpha }}{r}{\left(\frac{{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}}{{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}}\right)}^{\frac{r}{m-p}}{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}\right)}^{\frac{r}{p}}{c}_{\alpha ,r}^{-\frac{r}{p}}\\ \le s_0{g}_{\alpha }{\left[{\phi }_u\left(t_0\right)\right]}^{\frac{r}{p}}\end{array}$

其中 $s_0>0$ 独立于u， $g_{\alpha }={\Vert g\Vert }_{\infty }$。经计算可得

${\varnothing }_u\left(t_0\right)\ge {\phi }_u\left(t_0\right)-\lambda s_0{\phi }_u{\left(t_0\right)}^{\frac{r}{p}}\ge {\phi }_u{\left(t_0\right)}^{\frac{r}{p}}\left({\phi }_u{\left(t_0\right)}^{\frac{p-r}{p}}-\lambda s_0\right)$. (10)

则对任意 $u\in X\backslash \left\{0\right\}$，由于 ${\phi }_u\left(t_0\right)\ge \delta$，对任意非零u有 $\lambda <c_0^{-1}{\delta }^{\frac{p-r}{p}}={\lambda }_1$，进而可以得到函数 ${\varnothing }_u\left(t_0\right)>0$。证毕。

引理5假定(A₁)~(A₂)成立。如果 $0<\lambda <{\lambda }_1$，则存在 ${\delta }_1>0$ 使得对任意 $u\in M_{\lambda }^{-}\left(\Omega \right)$ 有 $J_{\lambda }\left(u_0\right)\ge {\delta }_1$。

证明：若 $u\in M_{\lambda }^{-}\left(\Omega \right)$，则 ${{\varnothing }^{″}}_u\left(1\right)<0$。函数 ${\varnothing }_u$ 在 $t=1$ 有一个正的局部极大值且 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$，由(10)可得

$J_{\lambda }\left(u\right)={\varnothing }_u\left(1\right)\ge {\varnothing }_u\left(t_0\right)\ge {\phi }_u{\left(t_0\right)}^{\frac{r}{p}}\left({\phi }_u{\left(t_0\right)}^{\frac{p-r}{p}}-\lambda s_0\right)\ge {\delta }^{\frac{r}{p}}\left({\delta }^{\frac{p-r}{p}}-\lambda s_0\right)$.

引理6假设(A₁)~(A₂)且 $0<\lambda <{\lambda }_1$ 成立，则对任意 $u\in M_{\lambda }\left(\Omega \right)$，$u\ne 0$ 有

${\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}\ne 0$,

即 $M_{\lambda }^0\left(\Omega \right)=\varnothing$。

证明：假设结果不成立，则存在 $u\in M_{\lambda }^0\left(\Omega \right)\subset M_{\lambda }\left(\Omega \right)$ 有

${\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}=\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}$. (11)

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$，则有

${{\varnothing }^{″}}_u\left(1\right)=\left(p-r\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}+\left(r-m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$,

从而 $u\notin M_{\lambda }^0$。

若 ${\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$。由(11)可得

${{\varnothing }^{″}}_u\left(1\right)=\left(p-m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}+\left(p-r\right){\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}=0$.

进而有

$-{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}=\frac{p-m}{p-r}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}$. (12)

此外

${{\varnothing }^{″}}_u\left(1\right)=\left(p-r\right){\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}+\left(r-m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}=0$.

从而

${\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}=\frac{m-r}{p-r}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}>0$. (13)

由(12)和(13)可得

$\begin{array}{c}{\varnothing }_u\left(t\right)=\frac{1}{p}{t}^p{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^p\text{d}x}-\frac{1}{m}{t}^m{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-\frac{1}{r}{t}^r{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u\right|}^r\text{d}x}\\ =\frac{1}{p}{t}^p\frac{r-m}{p-r}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-\frac{1}{m}{t}^m{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}-\frac{1}{r}{t}^r\frac{p-m}{p-r}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\\ =\left(\frac{m-r}{p\left(p-r\right)}{t}^p+\frac{p-m}{r\left(p-r\right)}{t}^r-\frac{1}{m}{t}^m\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}.\end{array}$

特别地，由假设条件可得到

${\varnothing }_u\left(t_0\right)=\frac{\left(m-p\right)t_0^p}{p-r}\left(\frac{m-r}{pm}-\frac{1}{r}{\left(\frac{p-r}{m-r}\right)}^{\frac{p-r}{m-p}}\right){\displaystyle {\int }_{\Omega }h\left(x\right){\left|u\right|}^m\text{d}x}\le 0$.

由引理4可知这是一个矛盾。从而有 $M_{\lambda }^0\left(\Omega \right)=\varnothing$。证毕。

3. 正解的存在性

由引理6可知当 $0<\lambda <{\lambda }_1$ 时 $M_{\lambda }\left(\Omega \right)=M_{\lambda }^{+}\left(\Omega \right)\cup M_{\lambda }^{-}\left(\Omega \right)$。

引理7假定(A₁)~(A₂)成立。如果 $0<\lambda <{\lambda }_1$，则函数 $J_{\lambda }$ 在 $M_{\lambda }^{+}\left(\Omega \right)$ 上有一个极小值。

证明：由引理1可得泛函 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }\left(\Omega \right)$ 上有下界，则泛函 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }^{+}\left(\Omega \right)$ 上也有下界。因此存在一个极小值序列 $\left\{u_k\right\}\subset M_{\lambda }^{+}\left(\Omega \right)$，有

${\lim }_{k\to \infty }{J}_{\lambda }\left(u_k\right)={\inf }_{u\in M_{\lambda }^{+}\left(\Omega \right)}{J}_{\lambda }\left(u\right)$.

由于泛函 $J_{\lambda }\left(u\right)$ 是强制的，序列 $\left\{u_k\right\}$ 在空间X上有界。在不失一般性的条件下在空间X中有 $u_k$ 弱收敛于u。对 $1<q<\frac{Np}{N-p}$，在 $L^q\left(\Omega \right)$ 中有 $u_k\to u_0$。

选取 $u\in X$ 使得 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u\right|}^r\text{d}x}>0$，存在 $t_1\left(u\right)$ 使得 $t_1\left(u\right)\in M_{\lambda }^{+}\left(\Omega \right)$ 和 $J_{\lambda }\left(t_1\left(u\right)u\right)<0$。则有 ${\inf }_{u\in M_{\lambda }^{+}\left(\Omega \right)}{J}_{\lambda }\left(u\right)<0$ 和 $J_{\lambda }\left(u_k\right)<0$。

由(3)可得

$J_{\lambda }\left(u_k\right)=\left(\frac{1}{p}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u_k\right|}^p\text{d}x}-\left(\frac{1}{r}-\frac{1}{m}\right)\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u_k\right|}^r\text{d}x}$.

进而可得到

$\left(\frac{1}{r}-\frac{1}{m}\right)\lambda {\displaystyle {\int }_{\Omega }g\left(x\right){\left|u_k\right|}^r\text{d}x}=\left(\frac{1}{p}-\frac{1}{m}\right){\displaystyle {\int }_{\Omega }{\left|\nabla u_k\right|}^p\text{d}x}-J_{\lambda }\left(u_k\right)$.

令 $k\to \infty$，有 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u_0\right|}^r\text{d}x}>0$。

假设在空间X中 $u_k$ 不收敛于u，而由纤维映射 ${\varnothing }_{u_0}\left(t\right)$ 则得到矛盾。由于 ${\displaystyle {\int }_{\Omega }g\left(x\right){\left|u_0\right|}^r\text{d}x}>0$，则存在 $t_0>0$ 使得 $t_0{u}_0\in M_{\lambda }^{+}\left(\Omega \right)$ 且 ${\varnothing }_{u_0}\left(t\right)$ 在 $\left(0,t_0\right)$ 上单调递减， ${{\varnothing }^{\prime }}_{u_0}\left(t_0\right)=0$。由于在空间X中 $u_k$ 不收敛于u，从而可以得到

${\displaystyle {\int }_{\Omega }{\left|\nabla u_k\right|}^p\text{d}x}<\underset{k\to \infty }{\lim \inf }{\displaystyle {\int }_{\Omega }{\left|\nabla u_k\right|}^p\text{d}x}$.

由于序列 $\left\{u_k\right\}\subset M_{\lambda }^{+}\left(\Omega \right)$，从而

${{\varnothing }^{\prime }}_{u_k}\left(t\right)=t^{p-1}{\displaystyle {\int }_{\Omega }{\left|\nabla u_k\right|}^p\text{d}x}-t^{m-1}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u_k\right|}^m\text{d}x}-t^{r-1}{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u_k\right|}^r\text{d}x}$.

且

${{\varnothing }^{\prime }}_{u_0}\left(t\right)=t^{p-1}{\displaystyle {\int }_{\Omega }{\left|\nabla u_0\right|}^p\text{d}x}-t^{m-1}{\displaystyle {\int }_{\Omega }h\left(x\right){\left|u_0\right|}^m\text{d}x}-t^{r-1}{\displaystyle {\int }_{\Omega }\lambda g\left(x\right){\left|u_0\right|}^r\text{d}x}$.

则由 ${{\varnothing }^{\prime }}_{u_0}\left(t_0\right)=0$ 可得

$\begin{array}{c}{{\varnothing }^{\prime }}_{u_k}\left(t_0\right)=t_0^{p-1}{\displaystyle {\int }_{\Omega }\left({\left|\nabla u_k\right|}^p-{\left|\nabla u_0\right|}^p\right)\text{d}x}-t_0^{m-1}{\displaystyle {\int }_{\Omega }h\left(x\right)\left({\left|u_k\right|}^m-{\left|u_0\right|}^m\right)\text{d}x}\\ -t_0^{r-1}{\displaystyle {\int }_{\Omega }\lambda g\left(x\right)\left({\left|u_k\right|}^r-{\left|u_0\right|}^r\right)\text{d}x}.\end{array}$

由于在 $L^m\left(\Omega \right)$ 中 $u_k\to u_0$，从而

$\underset{k\to \infty }{\lim \inf }{{\varnothing }^{\prime }}_{u_0}\left(t_0\right)\ge t_0^{p-1}{\displaystyle {\int }_{\Omega }\left({\left|\nabla u_k\right|}^p-{\left|\nabla u_0\right|}^p\right)\text{d}x}>0$.

这表明当k充分大时，有 ${{\varnothing }^{\prime }}_{u_k}\left(t_0\right)>0$。由于 $\left\{u_k\right\}\subset M_{\lambda }^{+}\left(\Omega \right)$，即 ${{\varnothing }^{″}}_{u_k}\left(1\right)>0$，易看出对 $0<t<1$ 有 ${{\varnothing }^{\prime }}_{u_k}\left(t\right)<0$ 且对任意k有 ${{\varnothing }^{\prime }}_{u_k}\left(1\right)=0$。从而可以得到 $t_0>1$。由于 $t_0{u}_0\in M_{\lambda }^{+}\left(\Omega \right)$ 和 ${{\varnothing }^{\prime }}_{u_0}\left(t_0\right)=0$，则 ${\varnothing }_{u_0}\left(t_0\right)$ 是一个局部极小值且 ${\varnothing }_{u_0}\left(t_0\right)<{\varnothing }_{u_0}\left(1\right)$。因此有

$J_{\lambda }\left(t_0{u}_0\right)<J_{\lambda }\left(u_0\right)<\underset{k\to \infty }{\lim }{J}_{\lambda }\left(u_k\right)=\underset{u\in M_{\lambda }^{+}\left(\Omega \right)}{\inf }{J}_{\lambda }\left(u\right)$.

矛盾！则在空间X中有 $u_k\to u_0$，从而

$J_{\lambda }\left(u_0\right)=\underset{k\to \infty }{\lim }{J}_{\lambda }\left(u_k\right)=\underset{u\in M_{\lambda }^{+}\left(\Omega \right)}{\inf }{J}_{\lambda }\left(u\right)$.

因此 $u_0$ 是泛函 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }^{+}\left(\Omega \right)$ 中的一个极小值点。

引理8假设(A₁)~(A₂)且 $0<\lambda <{\lambda }_1$，则函数 $J_{\lambda }\left(u\right)$ 在集合 $M_{\lambda }^{-}\left(\Omega \right)$ 上有一个极小值。

证明：由引理5可得存在 ${\delta }_1>0$，对任意 $u\in M_{\lambda }^{-}\left(\Omega \right)$ 有 $J_{\lambda }\left(u\right)\ge {\delta }_1>0$，则有 ${\inf }_{u\in M_{\lambda }^{+}\left(\Omega \right)}{J}_{\lambda }\left(u\right)\ge {\delta }_1$。因此存在一个极小值序列 $\left\{u_k\right\}\subset M_{\lambda }^{-}\left(\Omega \right)$ 使得