A priori bounds for positive solutions of a semilinear elliptic equation

Abstract:

We consider the semilinear elliptic equation $- \Delta u = f(u)$, $x \in \Omega$, subject to zero Dirichlet boundary conditions, where $\Omega \subset {{\mathbf {R}}^n}$ is a bounded domain with smooth boundary and the nonlinearity $f$ assumes both positive and negative values. Under the assumption that $\Omega$ satisfies certain symmetry conditions we establish two results providing lower bounds on the ${C^0}(\overline \Omega )$ norm of positive solutions. The bounds derived are the same one obtains in dimension $n = 1$.