A coupled Schrödinger system with critical exponent

Abstract

For the coupled Schrödinger system

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V_1(x)u=\mu _1u^3+\beta uv^2\ \ \text{ in }\ {{\mathbb {R}}}^4,\\&-\Delta v+V_2(x)v=\beta u^2v+\mu _2v^3\ \ \,\text{ in }\ {{\mathbb {R}}}^4,\\&\ u\ge 0,\ v\ge 0\ \ \text{ in }\ {{\mathbb {R}}}^4,\ \ u, v\in D^{1,2}({{\mathbb {R}}}^4), \end{aligned}\right. \end{aligned}$$

where $V_1, V_2\in L^2({{\mathbb {R}}}^4)\,\cap \, L_\mathrm{loc}^\infty ({{\mathbb {R}}}^4)$ are nonnegative functions and $\mu _1, \mu _2, \beta$ are positive constants, we prove that if $\beta >\max \{\mu _1, \mu _2\}$, $|V_1|_{L^2({{\mathbb {R}}}^4)}+|V_2|_{L^2({{\mathbb {R}}}^4)}>0$, and $|V_1|_{L^2({{\mathbb {R}}}^4)}$ and $|V_2|_{L^2({{\mathbb {R}}}^4)}$ are suitably small, a positive solution exists. This generalizes the well known result in [8] by Benci and Cerami on a scalar equation to the above system.