Positive solutions of elliptic systems with superlinear terms on the critical hyperbola (Q6655901)

Let \(\Omega\subset\mathbb{R}^N\), \(N\geq 3\) be a bounded domain of class \(C^2\). The authors discuss the existence of positive solutions to the system \N\[ \N\begin{cases} -\Delta v=\frac{u^p}{\ln^\alpha(e+u)}&\quad\mbox{ in }\Omega,\\\N-\Delta u = \frac{v^q}{\ln^\beta(e+v)}&\quad\mbox{ in }\Omega,\\\Nu=v=0&\quad\mbox{ on }\partial\Omega,\\\N\end{cases} \N\] \Nwhere \(p, q>0\), \(\alpha\leq p\), \(\beta\leq q\) and either \N\[\N 1>\frac{1}{p+1}+\frac{1}{q+1}>\frac{N-2}{N} \N\] \Nor \N\[\N \frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}\quad\mbox{ and }\quad \frac{\alpha}{p+1}+\frac{\beta}{q+1}>0. \N\]\NThe main result of the article establishes the existence of a positive mountain pass type solution. The approach relies on the dual method which requires the authors to investigate the critical points of an energy functional defined on an appropriate Orlicz space.