Nonlinear boundary value problems for second order elliptic systems (Q1059780)

The author treats boundary value problems for semilinear weakly coupled elliptic systems of second order \({\mathcal L}u_ r=f_ r(x,u,\partial u)\), \(x\in \Omega\), \(r=1,...,N\); \(B_ ru_ r=\Phi_ r\), \(x\in \partial \Omega\). Here \({\mathcal L}\) is a scalar, linear elliptic differential operator, the right hand sides \(f_ r\) are of weaker than quadratic growth with respect to \(\partial u\), and \(B_ r\) stands for either the Dirichlet or the ''third'' boundary condition. Employing an inclusion principle which generalizes the usual concept of sub- and supersolutions for scalar equations the author proves the existence of a solution to the above problem. The necessary a-priori estimates are rather easy since the nonlinear terms are of subquadratic growth.