Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth

Abstract

In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: \begin{equation*} \left\{ \begin{array}{ll} (-Δ+m^{2})^{s}u + V(\varepsilon x) u= f(u)+u^{2^{*}_{s}-1} \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, $s\in (0,1)$, $m>0$, $N>2s$, $2_s^{\ast }=\frac{2N}{N-2s}$ is the fractional critical exponent, $(-\Delta +m^2{)}^s$ is the fractional relativistic Schrödinger operator, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential, and $f:\mathbb{R}\to \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential $V$, we construct a family of positive solutions $u_{\epsilon }\in H^s({\mathbb{R}}^N)$, with exponential decay, which concentrates around a local minimum of $V$ as $\epsilon \to 0$.

arXiv commentThis is the final version (several modifications have been made with respect to the previous versions)

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