A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems. (English) Zbl 0823.35064

The author deals with the existence of positive solutions for the system \(-\Delta u= M(x)u+ N(x,u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Omega \subset \mathbb{R}^ N\), \(N\geq 2\), is a regular bounded domain, \(u= (u_ 1, u_ 2)\), \(M(x)= (m_{ij} (x))\) is a \(2\times 2\) matrix with \(m_{ij} (x)\geq 0\), for all \(x\in \Omega\), \(i\neq j\) and \(N(x,u)= (f(x,u), g(x,u))\) when \(f\) has asymptotic behavior at \(+\infty\) as \(u^ \sigma_ 1\) and \(g\) satisfies some subcritical growth. The proofs are based on a known fixed point theorem in conical shells and the a priori bounds technique.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J50 Variational methods for elliptic systems
47J05 Equations involving nonlinear operators (general)