Sign-changing solutions to Schrödinger-Kirchhoff-type equations with critical exponent

Advances in Difference Equations

Liping Xu¹  Haibo Chen² 

[1] Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, P.R. China;School of Mathematics and Statistics, Central South University, Changsha, P.R. China

关键词: Schrödinger-Kirchhoff-type equations;    critical nonlinearity;    positive solutions;    sign-changing solutions;    variational methods;    35J20;    35J65;    35J60;   

DOI  :  10.1186/s13662-016-0828-0

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we study the following Schrödinger-Kirchhoff-type equations:{−(a+b∫R3|∇u|2dx)△u+u=k(x)|u|2∗−2u+μh(x)uin R3,u∈H1(R3),$$ \textstyle\begin{cases}-(a+b\int_{\mathrm{R}^{3}}|\nabla u|^{2}\,dx)\triangle u+u= k(x)|u|^{2^{*}-2}u+\mu h(x)u \quad\mbox{in } \mathrm{R}^{3},\\ u\in H^{1}(\mathrm{R}^{3}), \end{cases} $$wherea,b,μ>0$a, b,\mu>0$are constants,2∗=6$2^{*}=6$is the critical Sobolev exponent in three spatial dimensions. Under appropriate assumptions on nonnegative functionsk(x)$k(x)$andh(x)$h(x)$, we establish the existence of positive and sign-changing solutions by variational methods.

【 授权许可】

CC BY