带有Hardy项的变指数椭圆型方程变号解的存在性  

THE EXISTENCE OF SIGN-CHANGING SOLUTIONS FOR VARIABLE EXPONENTIAL ELLIPTIC EQUATIONS WITH HARDY TERM

作　　者：范凤文 王琳琳 Fan Fengwen;Wang Linlin(School of Mathematics and Statistics Sciences,Ludong University,264039,Yantai,Shandong,China)

机构地区：[1]鲁东大学数学与统计科学学院,山东烟台264025

出　　处：《山东师范大学学报（自然科学版）》2022年第4期335-344,共10页Journal of Shandong Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(11201213)。

摘　　要：本文主要研究如下带有Hardy项的变指数半线性椭圆型方程-Δu-γu|x|^(2)=|u|^(q(x)-2)u+λu,x∈Ω\{0},u=0,x∈∂Ω,变号解的存在性,其中Ω⊂ℝ^(N)(N≥3)是有界光滑区域,0≤γ<γ-:=((N-2/2)^(2),q(x)是Ω上的连续函数,0<λ<λ_(1)(γ).首先对原方程的非线性项进行修改,得到一个新的摄动方程;其次利用变分方法对新的摄动方程进行研究,得到了摄动方程有无穷多个变号解;最后通过对变号解进行估计证明了摄动方程的解就是原方程的解,从而可知原方程有无穷多个变号解.This paper is devoted to study the existence of sign-changing solutions to the following variable exponential semilinear elliptic problem.-Δu-γu|x|^(2)=|u|^(q(x)-2)u+λu,x∈Ω\{0},u=0,x∈∂Ω where Ω⊂ℝ^(N)(N≥3) is a bounded smooth domain,0<λ<λ_(1)(γ)and 0≤γ<γ-:=((N-2/2)^(2),q(x)is a continuous function in Ω.Firstly,a new perturbation equation is obtained by changing the original equation.Secondly,the solutions of the perturbation equation are studied by means of the method of variation.Finally,the solutions of the perturbation equation are proved to be the solutions of the original equation by a estimation.Thus,the existence of infinitely many variation solutions to the original equation is obtained.

关 键 词：半线性椭圆型方程 变分方法 变号解 

分 类 号：O175.2[理学—数学]