Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation

Abstract

In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-Δ)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\epsilon >0$ is a small parameter, $s\in (\frac{3}{4},1)$, $(-\Delta {)}^s$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential and $f$ is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.

arXiv commentMathematical Methods in the Applied Sciences (2017)

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• Fractional Quantum Mechanics and Levy Path Integrals
• Nonlocal diffusion and applications
• Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$
• Multiplicity results for some nonlinear Schroedinger equations with potentials
• Multiplicity and concentration behavior of positive solutions for a Schrodinger-Kirchhoff type problem via penalization method
• Concentration phenomenon for fractional nonlinear Schrödinger equations

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• Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth
• Nodal solutions for double phase Kirchhoff problems with vanishing potentials
• Existence, multiplicity and concentration for a class of fractional $p\&q$ Laplacian problems in ${\mathbb{R}}^N$
• Multiplicity of solutions for fractional Schrödinger systems in ${\mathbb{R}}^N$
• Infinitely many periodic solutions for a class of fractional Kirchhoff problems
• On a fractional magnetic Schrödinger equation in $\mathbb{R}$ with exponential critical growth