Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth

Source abstract

&lt;abstract&gt;&lt;p&gt;In this paper, we deal with the existence of nontrivial solutions to the following class of strongly coupled Hamiltonian systems:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \quad \left\{ \begin{array}{rclll} -{\rm div} \big(w(x)\nabla u\big) \ = \ g(x,v),&amp;amp;\ &amp;amp; x \in B_1(0), \\[5pt] - {\rm div}\big(w(x) \nabla v\big)\ = \ f(x,u),&amp;amp;\ &amp;amp; x \in B_1(0), \\[5pt] u = v = 0&amp;amp;\ &amp;amp; x \in \partial B_1(0), \end{array} \right. \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;