On the periodic and asymptotically periodic nonlinear Helmholtz equation

Gilles Evequoz

Published 2015-10-28Version 1

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven under the assumptions $N\geq 3$, $\lambda>0$, $Q\in L^\infty(\mathbb{R}^N)$, $\mathbb{Z}^N$-periodic, $Q\geq 0$, $Q\not\equiv 0$ and for $p$ in the subcritical range $\frac{2(N+1)}{N-1}<p<\frac{2N}{N-2}$. In a second part, the existence of a nontrivial solution is shown in the case where the coefficient $Q$ is only asymptotically periodic.