Hamiltonian elliptic systems in $$\mathbb {R}^2$$ R 2 with subcritical and critical exponential growth

Abstract

Our aim in this paper was to establish by a variational method the existence of nontrivial solutions for strongly coupled Hamiltonian systems of the form $$\begin{aligned} \left\{ \begin{array}{llll} -\Delta u +V(x) u&{} = g(x,v), &{}\quad v > 0 &{} \quad \text {in}\; \mathbb {R}^2, \\ -\Delta v +V(x) v&{} = f(x,u), &{}\quad u > 0 &{} \quad \text {in}\; \mathbb {R}^2, \end{array} \right. \end{aligned}$$when the potential \(V\) is neither bounded away from zero, nor bounded from above. The nonlinear terms \(g(x,s)\) and \(f(x,s)\) are superlinear at infinity and have exponential subcritical or critical growth of the Trudinger–Moser type. Typical features of this class of problems are the lack of compactness because of the unboundedness of the domain and the critical growth. Moreover, the Lagrangian functional associated with this class of systems is strongly indefinite, that is, it has a saddle-point geometry where both positive and negative subspaces of the quadratic form are infinite-dimensional. To overcome these difficulties, we use an inequality of Trudinger–Moser type combined with Galerkin methods and a linking theorem.