Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential

Abstract

Consider the Schrödinger--Bopp--Podolsky system \[ \begin{cases} -ε^2Δu+(V+Kϕ)u=u|u|^{p-1};\newline Δ^2ϕ-Δϕ=4πK u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $\epsilon >0$, where $V,K:{\mathbb{R}}^3\to [0,\infty [$; $p\in ]1,5[$ are fixed and we want to solve for $u,\varphi :{\mathbb{R}}^3\to \mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$.

22 pages; comments are welcome

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