On a class of Hamiltonian systems with Trudinger-Moser nonlinearities

Abstract

In the present paper, we study existence of nontrivial weak solutions for a class of Hamiltonian systems of type \begin{align} \left \{ \begin{array}{ll} -\Delta u +b(x) u = w_{1}(x)f_{1}(v), & v > 0; \\ -\Delta v +b(x) v =w_{2}(x)f_{2}(u), & u > 0 , \end{array} \right . \end{align} where $b:\mathbb{R}^2 \rightarrow \mathbb{R}$ is a continuous potential which may change sign and the nonlinearity $f_{i}:\mathbb{R} \rightarrow \mathbb{R}$ has critical or subcritical exponential growth in the sense of Trudinger-Moser's inequality, for $i=1,2$. The main results are proved by using variational methods through strongly indefinite functionals.