\(L^{\infty}\) bounds for solutions of parametrized elliptic equations (Q1071947)

The author considers a family of elliptic partial differential equations \(Au=g(\cdot,u,\lambda)\) where g is a positive function and A is a second order strongly elliptic operator in a bounded region \(\Omega \subset {\mathbb{R}}^ n\). Either Dirichlet or Neumann boundary conditions are assumed for u. By using maximum principle arguments it is shown that \(\| u\|_{\infty}\leq y(\lambda)\), where \(y(\lambda)\) is the solution of a certain ordinary differential equation related to the boundary conditions.