Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities

Abstract

In this paper we are concerned with the existence and multiplicity of solutions for the fractional Choquard-type Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity: \begin{eqnarray*} \begin{cases} \varepsilon^{2s}M([u]_{s,A}^2)(-Δ)_{A}^su + V(x)u = (|x|^{-α}*F(|u|^2))f(|u|^2)u + |u|^{2_s^\ast-2}u,\ \ \ x\in \mathbb{R}^N,\\ u(x) \rightarrow 0,\ \ \quad \mbox{as}\ |x| \rightarrow \infty, \end{cases} \end{eqnarray*} where $(-\Delta {)}_A^s$ is the fractional magnetic operator with $0<s<1$, $2_s^{\ast }=2N/(N-2s)$, $\alpha <\min \{N,4s\}$, $M:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+}$ is a continuous function, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is the magnetic potential, $F(|u|)={\int }_0^{|u|}f(t)dt$, and $\epsilon >0$ is a positive parameter. The electric potential $V\in C({\mathbb{R}}^N,{\mathbb{R}}_0^{+})$ satisfies $V(x)=0$ in some region of ${\mathbb{R}}^N$, which means that this is the critical frequency case. We first prove the $(PS{)}_c$ condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term $M$ can vanish at zero.

arXiv admin note: substantial text overlap with arXiv:1803.05694

References in corpus (6)

• Magnetic Pseudodifferential Operators
• Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials
• On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
• Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field
• On doubly nonlocal fractional elliptic equations
• Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

Cited by in corpus (3)

• On the $p$-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
• Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth
• On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields