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