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

Abstract

In this paper, we consider the following two-component elliptic system with critical growth \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ)u=μ_1u^{3}+βuv^{2}, \ \ x\in \mathbb{R}^4, -Δv+(V_2(x)+λ)v=μ_2v^{3}+βvu^{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 Schrödinger 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)).