Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth

Abstract

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\epsilon$ is a small positive parameter, $a,b>0$ are fixed constants, $s\in (\frac{3}{4},1)$, $2_s^{\ast }=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta {)}_A^s$ is the fractional magnetic Laplacian, $A:{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a smooth magnetic potential, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a positive continuous potential verifying the global condition due to Rabinowitz \cite{Rab}, and $f:\mathbb{R}\to \mathbb{R}$ is a $C^1$ subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions \cite{Lions}, a fractional Kato's type inequality and the Ljusternik-Schnirelmann theory of critical points.

arXiv admin note: text overlap with arXiv:1808.09295

References in corpus (5)

• A multiplicity result for a fractional Kirchhoff equation in ${\mathbb{R}}^N$ with a general nonlinearity
• On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity
• Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation
• Existence and concentration results for some fractional Schrödinger equations in ${\mathbb{R}}^N$ with magnetic fields
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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