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

In this paper we study the semilinear elliptic problem $-Δu -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2,$ where $k>0$, $p\geq 6$ 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.

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