In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - \Delta u - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{ \frac{ 1-N}{ 2}}) \quad \text{ and} \quad \frac{ \partial^2 u}{ \partial r^2}(x)+k^2 u(x)) =o(|x|^{ \frac{ 1-N}{ 2}}) \qquad \text{ as $r=|x| \to \infty$.} $$ 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.
Categories: math.AP