Multiplicity and concentration results for fractional Schrödinger-Poisson equations with magnetic fields and critical growth

Abstract

We deal with the following fractional Schrödinger-Poisson equation with magnetic field \begin{equation} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{2^{*}_{s}-2}u \quad \mbox{in} \mathbb{R}^{3}, \nonumber \end{equation} where $\epsilon >0$ is a small parameter, $s\in (\frac{3}{4},1)$, $t\in (0,1)$, $2_s^{\ast }=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta {)}_A^s$ is the fractional magnetic Laplacian, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a positive continuous potential, $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a smooth magnetic potential and $f:\mathbb{R}\to \mathbb{R}$ is a subcritical nonlinearity. Under a local condition on the potential $V$, we study multiplicity and concentration of nontrivial solutions as $\epsilon \to 0$. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential $V$ attains its minimum.

arXiv admin note: text overlap with arXiv:1801.00199, Potential Analysis (2018)

References in corpus (1)

• Singular perturbation by bending for an adhesive obstacle problem

Cited by in corpus (2)

• Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent
• On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields