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.

Mathematical Methods in the Applied Sciences (2017)