Concentration phenomena for a class of fractional Kirchhoff equations in ${\mathbb{R}}^N$ with general nonlinearities

Abstract

In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\epsilon >0$ is a small parameter, $s\in (0,1)$, $N\ge 2$, $(-\Delta {)}^s$ is the fractional Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a positive continuous function, $M:[0,\infty )\to \mathbb{R}$ is a Kirchhoff function satisfying suitable conditions and $f:\mathbb{R}\to \mathbb{R}$ fulfills Berestycki-Lions type assumptions of subcritical or critical type. Using suitable variational arguments, we prove the existence of a family of positive solutions $(u_{\epsilon })$ which concentrates at a local minimum of $V$ as $\epsilon \to 0$.

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