Existence of least energy sign-changing solution for the nonlinear Schrödinger system with two types of nonlocal terms

Boundary value problems

Yuhua Li¹  Yawen Hao¹ 

[1] School of Mathematical Sciences, Shanxi University, Taiyuan, P.R. China

关键词: Schrödinger-Poisson system;    Kirchhoff;    sign-changing solution;    Brouwer’s degree;    constraint variational;    47J30;    34B15;   

DOI  :  10.1186/s13661-016-0728-y

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In the paper, we are concerned with the system of Kirchhoff-Schrödinger-Poisson system under certain assumptions onV1$V_{1}$,V2$V_{2}$, K and f. We are interested in the existence of least energy sign-changing solutions to the system onRN$\mathbb{R}^{N}$. Because two kinds of nonlocal termsϕu$\phi_{u}$and∫RN|∇u|2$\int_{\mathbb{R}^{N}}|\nabla u|^{2}$are involved in the system, the methods are different from the Kirchhoff or the Schrödinger-Poisson system. The two nonlocal terms∫RN|∇u|2$\int_{\mathbb{R}^{N}}|\nabla u|^{2}$andϕu$\phi_{u}$make that the functionalJ(u++u−)≠J(u+)+J(u−)$J(u^{+}+u^{-})\neq J(u^{+})+J(u^{-})$. Moreover, the nonlocal termϕu$\phi_{u}$does not have the convergence property because of the assumptionV2$V_{2}$. In addition, the convergence of these two nonlocal terms are different. In the present paper, we unify the increasing property conditions on sign-changing solution in previous papers. We construct a new homotopy operator and then weaken the assumption that f isC1$C^{1}$to that of f being only continuous. We prove that the system has a sign-changing solution via a constraint variational method combining with Brouwer’s degree theory.

【 授权许可】

CC BY