Ground State and Symmetry Breaking of Schr¨odinger-poisson-slatter Equations

Abstract

We study the nonlocal Schr\"{o}dinger-Poisson-Slater type equation $$-\Delta u+\omega u+\lambda(I_\alpha\star |u|^q)|u|^{q-2} u=|u|^{p-2}u, $$ where $u \in \dot{H^1}(\mathbb{R}^N) \cap L^p(\mathbb{R}^N), p>1, \omega \in \mathbb{R}, I_\alpha$ is the Riesz transform, and $q \geq 1$. Exploring the equation based on the sign of the real number $\omega$ reveals varied existence outcomes. We establish the existence of global minimizers for $\omega =1$ or $\omega = 0$, depending on the intervals where the generalized Gagliardo-Nirenberg inequalities hold for powers $p$ and $q$. Moreover, symmetry-breaking occurs within large balls, signifying the presence of both radial and non-radial minimizers for the equation in the ball. Additionally, we prove the existence of groundstates and provide a proof of their orbital stability.