On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

Abstract

We consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity ${\epsilon }^{2s}M([u{]}_{s,A_{\epsilon }}^2)(-\Delta {)}_{A_{\epsilon }}^{s}u+V(x)u=$ $|u{|}^{2_s^{\ast }-2}u+h(x,|u{|}^2)u,$ $x\in {\mathbb{R}}^N,$ where $u(x)\to 0$ as $|x|\to \infty ,$ and $(-\Delta {)}_{A_{\epsilon }}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^{\ast }=2N/(N-2s),$ $M:{\mathbb{R}}_0^{+}\to {\mathbb{R}}^{+}$ is a continuous nondecreasing function, $V:{\mathbb{R}}^N\to {\mathbb{R}}_0^{+},$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that $\epsilon <\mathcal{E}$; and (ii) for any $m^{\ast }\in \mathbb{N}$, has $m^{\ast }$ pairs of solutions if $\epsilon <{\mathcal{E}}_{m^{\ast }}$, where $\mathcal{E}$ and ${\mathcal{E}}_{m^{\ast }}$ are sufficiently small positive numbers. Moreover, these solutions $u_{\epsilon }\to 0$ as $\epsilon \to 0$.

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Cited by in corpus (8)

• Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities
• Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
• Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth
• Maz'ya-Shaposhnikova formula in Magnetic Fractional Orlicz-Sobolev spaces
• 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
• Characterization of the traces on the boundary of functions in magnetic Sobolev spaces
• On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields