Positive Solutions For a Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$

Abstract

We consider the following Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$ $$\{\begin{array}{c}-{\epsilon }^2\Delta u+V(x)u+\varphi u=f(u)\\ -{\epsilon }^2\Delta \varphi +{\epsilon }^4{\Delta }^2\varphi =4\pi \epsilon u^2\end{array}$$ where $\epsilon >0$ with $V:{\mathbb{R}}^3\to \mathbb{R},f:\mathbb{R}\to \mathbb{R}$ satisfy suitable assumptions. By using variational methods, we prove that the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of $M$, the set of minima of the potential $V$.

Cited by in corpus (2)

• Minimizers of mass-constrained functionals involving a nonattractive point interaction
• Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system