  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.
