Existence of positive solutions for a nonvariational quasilinear elliptic system
Let \(\Omega\) be an open ball with center \(0\) in \({\mathbb R}^N\) and \(\Delta_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)\) be the \(p\)-Laplacian of \(u\). The authors establish the existence of a positive radial solution of the boundary value problem \[ -\Delta u= u^\alpha u^\beta\quad\text{on }\Omega,\qquad -\Delta v= u^\gamma v^\delta\quad\text{on }\Omega,\qquad u,v\equiv 0\quad\text{on }\partial\Omega \] under, e.g., the following hypotheses: \(1<p\), \(q<N\), \(\alpha, \beta, \gamma, \delta\geq 0\), \(\beta\gamma>D:=(p-1-\alpha)(q-1-\delta)\), \(\min\{p-1-\alpha,q-1-\delta\}>0\), \(\max\{\sigma_1-{{N-p}\over{p-1}},\sigma_2-{{N-q}\over{q-1}}\}\geq 0\). Here \(\sigma_1:=[\beta q-p(q-1-\delta)]/D\) and \(\sigma_2:=[\gamma p -q(p-1-\alpha)]/D\). These results generalize earlier work by \textit{Ph. Clément, R. Manásevich} and \textit{E. Mitidieri} [Commun. Partial Differ. Equations 18, No. 12, 2071-2106 (1993; Zbl 0802.35044)]. A priori bounds are obtained by a blow-up method already used in this paper, and a degree argument is then applied. Also, some nonexistence results for the ground state are derived in case that the above system is considered on \({\mathbb R}^N\).
systems involving \(p\)-Laplacian without variational structure
scientific article; zbMATH DE number 1530280