Standing waves for two-component elliptic system with critical growth in $\mathbb{R}^{4}$: the attractive case

TL;DR

This paper establishes the existence and multiplicity of positive standing wave solutions for a two-component elliptic system with critical growth in four-dimensional space, using variational methods and degree theory.

Contribution

It extends previous results on Schrödinger equations to a coupled elliptic system with critical growth, providing new existence and multiplicity results under specific conditions.

Findings

Proved existence of positive solutions for the system.

Established multiplicity results under certain parameter conditions.

Generalized previous scalar results to a coupled system.

Abstract

In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -\Delta u+(V_1(x)+\lambda)u=\mu_1u^{3}+\beta uv^{2}, \ \ x\in \mathbb{R}^4, -\Delta v+(V_2(x)+\lambda)v=\mu_2v^{3}+\beta vu^{2}, \ \ x\in \mathbb{R}^4 , % u\geq 0, \ \ v\geq 0 \ \text{in} \ \R^4. \end{cases} \end{equation*} where $V_j(x) \in L^{2}(\mathbb{R}^4)$ are nonnegative potentials and the nonlinear coefficients $\beta ,\mu_j$, $j=1,2$, are positive. Here we also assume $\lambda>0$. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of positive solutions under the hypothesis $\beta>\max\{\mu_1,\mu_2\}$. These results generalize the results for semilinear Schr\"{o}dinger equation on half space by Cerami and Passaseo (SIAM J. Math. Anal., 28, 867-885, (1997)) to the above elliptic system, while extending the existence result from Liu and Liu (Calc. Var. Partial Differential Equations, 59:145, (2020)).