On Concentration of Real Solutions for Fractional Helmholtz Equation

Abstract

This paper studies the nonlinear fractional Helmholtz equation

$(-\Delta)^su-k^2u=Q(x)|u|^{p-2}u,\quad\text{in }\mathbb{R}^N,\,N\geq3,\tag{0.1}$

where $\frac{N}{N+1}<s<\frac{N}{2},\quad\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2s}$ are two real exponents, and the coefficient $Q(x)$ is a bounded continuous, nonnegative function that satisfies the condition

$\limsup\limits_{|x|\to\infty} Q(x) < \sup\limits_{x\in\mathbb{R}^N} Q(x). \tag{0.2}$

For sufficiently large $k>0$, the existence of real-valued solutions to $(0.1)$ is established. Furthermore, as $k\rightarrow∞$, it is shown that the sequence of solutions associated with the ground states of a dual equation concentrates, after rescaling, at global maximum points of the function $Q(x).$