Calculus of Variations and Geometric Measure Theory

H. Hajaiej - R. Kumar - T. Mukherjee - L. Song

Existence and non-existence results to a mixed Schrodinger system in a plane

created by kumar2 on 03 Feb 2024



Inserted: 3 feb 2024
Last Updated: 3 feb 2024

Year: 2023

ArXiv: 2311.16547 PDF


This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type \begin{equation} \left\{ \begin{aligned} -\partial{xx}u + (-\Delta){y}{s{1}} u + u - u{2{s{1}}{}-1} = \kappa \alpha h(x,y) u{\alpha-1}v{\beta} & \quad \mbox{in} ~ \mathbb{R}{2}, -\partial{xx}v + (-\Delta){y}{s{2}} v + v- v{2{s{2}}{}-1} = \kappa \beta h(x,y) u{\alpha}v{\beta-1} & \quad \mbox{in} ~ \mathbb{R}{2}, u,v ~ \geq ~0 \quad \mbox{in} ~ \mathbb{R}{2}, \end{aligned} \right. \end{equation} where $s_{1},s_{2} \in (0,1),~\alpha,\beta>1,~\alpha+\beta \leq \min \{ 2_{s_{1}}^{},2_{s_{2}}^{}\}$, and $2_{s_i}^{} = \frac{2(1+s_i)}{1-s_i}, i=1,2$. The existence of a ground state solution entirely depends on the behaviour of the parameter $\kappa>0$ and on the function $h$. In this article, we prove that a ground state solution exists in the subcritical case if $\kappa$ is large enough and $h$ satisfies (1.3). Further, if $\kappa$ becomes very small in this case then there does not exist any solution to our system. The study in the critical case, i.e. $s_1=s_2=s, \alpha+\beta=2_s$, is more complex and the solution exists only for large $\kappa$ and radial $h$ satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on $h$.