Real solutions to the nonlinear Helmholtz equation with local nonlinearity

Abstract

In this paper, we study real solutions of the nonlinear Helmholtz equation $$-\Delta u-k^{2}u=f(x,u),x\in {\mathbb{R}}^N$$ satisfying the asymptotic conditions $$u(x)=O(|x{|}^{\frac{1-N}{2}})\text{and}\frac{{\partial }^{2}u}{\partial r^2}(x)+k^{2}u(x))=o(|x{|}^{\frac{1-N}{2}})\text{as r=∣x∣→∞.}$$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.

Corrected version. To appear in Archive for Rational Mechanics and Analysis

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