Hamiltonian elliptic systems with nonlinearities of arbitrary growth

Abstract

We study the existence of standing wave solutions for the following class of elliptic Hamiltonian-type systems: \[ \begin{cases} -\hbar^2\Delta u+ V(x)u = g(v) & \mbox{in } \mathbb{R}^N, \\ -\hbar^2\Delta v+ V(x)v = f(u) & \mbox{in } \mathbb{R}^N, \end{cases} \] with $N\geq 2$, where $\hbar$ is a positive parameter and the nonlinearities $f,g$ are superlinear and can have arbitrary growth at infinity. This system is in variational form and the associated energy functional is strongly indefinite. Moreover, in view of unboundedness of the domain $\mathbb{R}^N$ and the arbitrary growth of nonlinearities we have lack of compactness. We use a dual variational approach in combination with a mountain-pass type procedure to prove the existence of positive solution for $\hbar$ sufficiently small.