Real solutions to an asymptotically linear Helmholtz equation

Abstract

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

Assuming that $f$ is asymptotically linear at infinity respect to $u$, the existence and multiplicity of real solutions are obtained via the variational methods.