On concentration of real solutions for fractional Helmholtz equation

Abstract

This paper studies the nonlinear fractional Helmholtz equation \begin{equation}\label{main} (-\Delta)^{s} u-k^{2} u=Q(x)|u|^{p-2}u, ~~\mathrm{in}~~\mathbb{R}^{N},~~N\geq3, \end{equation} where $\frac{N}{N+1}<s<\frac{N}{2}$, $\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ are two real exponents, and the coefficient $Q$ is bounded continuous, nonnegative and satisfies the condition \begin{equation} \mathop{\mathrm{lim~sup}}\limits_{|x|\longrightarrow\infty}Q(x) <\mathop{\mathrm{sup}}\limits_{x\in\mathbb{R}^{N}}Q(x). \end{equation} For $k>0$ large, the existence of real-valued solutions for (\ref{main}) are proved, and in the limit $k\longrightarrow\infty$, sequence of solutions associated with ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.