An existence result for gradient-type systems with a nondifferentiable term on unbounded strips
The author deals with the existence of nonzero solutions of the gradient-type system \[ -\Delta_p u= F_u(x,u,v)\quad\text{in }\Omega, \qquad -\Delta_q v= F_v(x,u,v)\quad\text{in }\Omega, \qquad u= v= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a strip-like domains of the form \(\Omega= \omega\times\mathbb R^{N-m}\) with \(\omega\subset\mathbb{R}^m\) \((m\geq 1)\) is an open bounded set, \(N-m\geq 2\). Here \(F\in C^1(\Omega\times \mathbb R^2, \mathbb R)\), \(1< p\), \(q< N\) with \(F_u\) is the partial derivative of \(F\) with respect to \(u\) (similarly for \(F_v\)) and \(\Delta_\alpha u:= \text{div}(|\nabla u|^{\alpha-2}\nabla u)\), \(\alpha\in \{p,q\}\). To this end, the author uses a nonsmooth version of the mountain pass theorem which involves the Cerami condition and principle of symmetric criticality for locally Lipschitz functions.
scientific article; zbMATH DE number 2141776
scientific article