## Multiple Positive Solutions for a Quasilinear Elliptic System Involving the Caffarelli-Kohn-Nirenberg Inequality and Nonlinear Boundary Conditions

Li Qin,1, Yang Zuodong,2

 基金资助: 安徽省自然科学基金青年项目.  1808085QA15安徽财经大学科学研究一般项目.  ACKYC19050国家自然科学基金.  11571093

Received: 2019-08-12

 Fund supported: the Anhui Natural Science Foundation.  1808085QA15the Scientific Research Project of Anhui University of Finance and Economics.  ACKYC19050the NSFC.  11571093

Abstract

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and nonlinear boundary conditions. By means of variational methods, the existence of at least two positive solutions is obtained.

Keywords： Quasilinear elliptic system ; Caffarelli-Kohn-Nirenberg inequality ; Nonlinear boundary conditions ; Nehari manifold

Li Qin, Yang Zuodong. Multiple Positive Solutions for a Quasilinear Elliptic System Involving the Caffarelli-Kohn-Nirenberg Inequality and Nonlinear Boundary Conditions. Acta Mathematica Scientia[J], 2020, 40(6): 1634-1645 doi:

## 1 引言

\begin{align} \left\{ \begin{array}{ll} { } -\mbox{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)-\mu \frac{|u|^{p-2}u}{|x|^{p(a+1)}} = \frac{2\alpha}{\alpha+\beta}\frac{|u|^{\alpha-2}u|v|^{\beta}}{|x|^{t}}, &x\in \Omega, \\ { } -\mbox{div}(|x|^{-ap}|\nabla v|^{p-2}\nabla v)-\mu \frac{|v|^{p-2}v}{|x|^{p(a+1)}} = \frac{2\beta}{\alpha+\beta}\frac{|u|^{\alpha}|v|^{\beta-2}v}{|x|^{t}}, &x\in \Omega, \\ { } |x|^{-ap}|\nabla u|^{p-2}\frac{\partial u}{\partial n} = \lambda f(x)|u|^{q-2}u, \;\ |x|^{-ap}|\nabla v|^{p-2}\frac{\partial v}{\partial n} = \delta g(x)|v|^{q-2}v, & x\in \partial\Omega \end{array}\right. \end{align}

\begin{align} \bigg(\int_{{{\Bbb R}} ^{N}}\frac{|u|^{p^{*}}}{|x|^{bp^{*}}}{\rm d}x\bigg)^{\frac{p}{p^{*}}}\leq K_{a, b}\int_{{{\Bbb R}} ^{N}}|x|^{-ap}|\nabla u|^{p}{\rm d}x, \;\;\ \forall u\in C_{0}^{\infty}({{\Bbb R}} ^{N}). \end{align}

\begin{align} \int_{{{\Bbb R}} ^{N}}\frac{|u|^{p}}{|x|^{p(a+1)}}{\rm d}x\leq \frac{1}{\overline{\mu}}\int_{{{\Bbb R}} ^{N}}|x|^{-ap}|\nabla u|^{p}{\rm d}x, \;\;\ \forall u\in C_{0}^{\infty}({{\Bbb R}} ^{N}), \end{align}

$\mu\in[0, \overline{\mu})$时, 令$E = W_{a}^{1, p}(\Omega)$表示$C_{0}^{\infty}(\Omega)$的闭包, 且赋有范数

\begin{align} \int_{\Omega}\frac{|u|^{p^{*}}}{|x|^{bp^{*}}}{\rm d}x\leq S^{-\frac{p^{*}}{p}}\|u\|^{p^{*}}. \end{align}

\begin{align} \left\{ \begin{array}{ll} { } -\mbox{div}(|x|^{-2a}\nabla u)-\mu \frac{u}{|x|^{2(1+a)}} = \frac{2\alpha}{\alpha+\beta}|x|^{-bp^{*}}|u|^{\alpha-2}u|v|^{\beta}+\lambda |u|^{q-2}u, &x\in \Omega, \\ { } -\mbox{div}(|x|^{-2a}\nabla v)-\mu \frac{v}{|x|^{2(1+a)}} = \frac{2\beta}{\alpha+\beta}|x|^{-bp^{*}}|u|^{\alpha}|v|^{\beta-2}v+\delta |v|^{q-2}v, &x\in \Omega, \\ u = v = 0, &x\in \partial\Omega, \end{array}\right. \end{align}

证明与文献[4]中的引理2.5类似, 故略去细节.

(ⅰ)若$K_{\lambda, \delta}(u, v)\leq0$, 则存在唯一的$s_{2}^{-}>\overline{s}_{0}$使得$(s_{2}^{-}u, s_{2}^{-}v)\in {\cal N}^{-}_{\lambda, \delta}$, $\phi_{u, v} $$(0, s_{2}^{-}) 上单调递增, 在 (s_{2}^{-}, +\infty) 上单调递减.并且 (ⅱ)若 K_{\lambda, \delta}(u, v)>0 , 则存在唯一的 0<s_{2}^{+}<\overline{s}_{0}<s_{2}^{-} 使得 (s_{2}^{+}u, s_{2}^{+}v)\in {\cal N}^{+}_{\lambda, \delta} , (s_{2}^{-}u, s_{2}^{-}v)\in {\cal N}^{-}_{\lambda, \delta} , \phi_{u, v}$$ (0, s_{2}^{+})$上单调递减, 在$(s_{2}^{+}, s_{2}^{-})$上单调递增, 在$(s_{2}^{-}, +\infty)$上单调递减.并且

## 3 定理1.1的证明

(ⅰ) $I_{\lambda, \delta}(u^{1}, v^{1}) = c_{\lambda, \delta}^{+}$;

(ⅱ) $(u^{1}, v^{1})$是问题(1.1)的一个正的弱解.

由于I_{\lambda, \delta} {\cal N}^{+}_{\lambda, \delta} 上下有界, 则存在极小化序列 \{(u_{n}, v_{n})\}\subset {\cal N}^{+}_{\lambda, \delta} 满足 \begin{align} \lim\limits_{n\rightarrow\infty}I_{\lambda, \delta}(u_{n}, v_{n}) = \inf\limits_{(u, v)\in {\cal N}^{+}_{\lambda, \delta}} I_{\lambda, \delta}(u, v) = c_{\lambda, \delta}^{+}<0. \end{align} 又由于 I_{\lambda, \delta} 是凸的, 易知 \{(u_{n}, v_{n})\} X中有界.从而存在子序列(仍记为$\{(u_{n}, v_{n})\}$)$(u^{1}, v^{1})\in X$使得当$n\rightarrow\infty$时, 有

\begin{align} \int_{\Omega}\frac{|u_{n}|^{\alpha}|v_{n}|^{\beta}}{|x|^{t}}{\rm d}x\rightarrow \int_{\Omega}\frac{|u^{1}|^{\alpha}|v^{1}|^{\beta}}{|x|^{t}}{\rm d}x, \;\ \mbox{当}\;\ n\rightarrow\infty, \end{align}

\begin{align} K_{\lambda, \delta}(u_{n}, v_{n})\rightarrow K_{\lambda, \delta}(u^{1}, v^{1}), \;\ \mbox{当}\;\ n\rightarrow\infty. \end{align}

\begin{align} \lim\limits_{n\rightarrow\infty}I_{\lambda, \delta}(u_{n}, v_{n}) = \inf\limits_{(u, v)\in {\cal N}^{-}_{\lambda, \delta}} I_{\lambda, \delta}(u, v) = c_{\lambda, \delta}^{-}. \end{align}

\begin{align} \int_{\Omega}\frac{|u^{2}|^{\alpha}|v^{2}|^{\beta}}{|x|^{t}}{\rm d}x>0. \end{align}

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