Elliptic Systems with a Partially Sublinear Local Term

Abstract

Let $1$$\begin{cases} -\Delta u+[V(x)-(\omega+e\phi)^2]u=K(x)|u|^{p-2}u \  \  \ {\rm in}  \ \mathbb{R}^3,\\ -\Delta\phi+e^2u^2\phi=-e\omega u^2 \  \  \ {\rm in} \ \mathbb{R}^3 \end{cases}$$ where $ω, e > 0.$ This is in sharp contrast to D'Aprile and Mugnai's non-existence results.