Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system

Abstract

Consider the following nonlinear Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$: $$\{\begin{array}{l}-{\epsilon }^2\Delta u+(V+\varphi )u=u|u{|}^{p-1};\\ a^2{\Delta }^2\varphi -\Delta \varphi =4\pi u^2,\end{array}$$ where $a,\epsilon >0$; $1<p<5$; $V:{\mathbb{R}}^3\to ]0,\infty [$ and we want to solve for $u,\varphi :{\mathbb{R}}^3\to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K\ge 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\epsilon$ is sufficiently small, then the system has a multipeak cluster solution with $K$ peaks placed at the vertices of a regular convex $K$-gon centered at $z_0$.

24 pages; major revision and different main result; comments are welcome

References in corpus (2)

• Positive Solutions For a Schrödinger-Bopp-Podolsky system in ${\mathbb{R}}^3$
• Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential