Ground state solutions to a class of critical Schrödinger problem

Advances in Nonlinear Analysis

article

Anmin Mao¹  Shuai Mo¹ 

[1] School of Mathematical Sciences, Qufu Normal University

关键词: Kirchhoff-Schrödinger equation;    Ground state;    Pohožaev identity;   

DOI  :  10.1515/anona-2020-0192

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2a+ε b∫ R3|∇ u|2Δ u+V(x)u=f(u)+u5,x∈ R3,u∈ H1(R3), $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$( SK ε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to ( SK ε ), where f ( u ) may behave like | u | q –2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V , we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V , we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

【 授权许可】

CC BY