A Priori Bounds for Positive Solutions of a Semilinear Elliptic Equation

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