Helmholtz equation with asymptotically linear nonlinearities

Abstract

In this paper, we investigate a quasilinear Helmholtz problem under Dirichlet boundary condition in a regular domain with asymptotically linear nonlinearities. We use variation method and Cerami version of the mountain pass theorem to prove the existence of solution without use the Ambrosetti-Rabionovitz condition or any of its refinements. Then, we prove that the same techniques work when the nonlinearity is superlinear and subcritical at infinity.