Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

Abstract

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$-\Delta u-k^{2}u=Q(x)|u{|}^{p-2}u,u\in W^{2,p}({\mathbb{R}}^N)$$ with $N\ge 3$, $\frac{2(N+1)}{(N-1)}<p<\frac{2N}{N-2}$ and nonnegative $Q\in L^{\infty }({\mathbb{R}}^N)$. We prove the existence of nontrivial solutions for periodic $Q$ as well as in the case where $Q(x)\to 0$ as $|x|\to \infty$. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.

References in corpus (1)

• Real solutions to the nonlinear Helmholtz equation with local nonlinearity

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