Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

Abstract

In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems: $ \begin{equation*} \quad \left\{ \begin{array}{rclll} -{\rm div} \big(w(x)\nabla u\big) \ = \ g(x,v),&\ & x \in B_1(0), \\[5pt] - {\rm div}\big(w(x) \nabla v\big)\ = \ f(x,u),&\ & x \in B_1(0), \\[5pt] u = v = 0&\ & x \in \partial B_1(0), \end{array} \right. \end{equation*} $ where $ w(x) = \big(\log 1/|x|\big)^{\gamma} $, $ 0\leq\gamma < 1 $, and the nonlinearities $ f $ and $ g $ possess exponential growth ranges above the exponential critical hyperbola. Our approach is based on Trudinger-Moser type inequalities for weighted Sobolev spaces and variational methods.

Keywords

• hamiltonian system
• trudinger-moser inequality
• supercritical exponential growth
• variational methods
• linking theorem