Real solutions to the nonlinear Helmholtz equation with local nonlinearity

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 $s a t i s f y i n g t h e a s y m p t o t i c c o n d i t i o n s$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad $and$ \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad $as r = ∣ x ∣ \to \infty .$ $$ 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.

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 $s a t i s f y i n g t h e a s y m p t o t i c c o n d i t i o n s$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad $and$ \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad $as r = ∣ x ∣ \to \infty .$ $$ 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.

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