On a class of critical elliptic systems in ℝ 4

Advances in Nonlinear Analysis

article

Xin Zhao¹  Wenming Zou¹ 

[1] Department of Mathematical Sciences, Tsinghua University

关键词: Schrödinger system;    Nehari manifold;    Ground state;   

DOI  :  10.1515/anona-2020-0136

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: − Δ u+λ 1u=μ 1u3+β uv2+2qpyu2qp− 1v2inΩ ,− Δ v+λ 2v=μ 2v3+β u2v+2yu2qpvinΩ ,u,v> -->0inΩ ,u,v=0on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ 4 is a smooth bounded domain with smooth boundary ∂Ω ; p , q are positive coprime integers with 1 0 and λ i ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

【 授权许可】

CC BY