Existence and asymptotic behavior of standing waves of the nonlinear Helmholtz equation in the plane

Abstract

In this paper we study the semilinear elliptic problem $$-\Delta u-k^{2}u=Q|u{|}^{p-2}u\text{ in }{\mathbb{R}}^2,$$ where $k>0$, $p\ge 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for decaying and for periodic coefficient $Q$. In addition, a nonlinear far-field relation is derived for these solutions.