A priori bounds for positive solutions of a semilinear elliptic equation

We consider the semilinear elliptic equation -△u = ƒ(u), x ∈ Ω, subject to zero Dirichlet boundary conditions, where ∈ ⊂ Rⁿ is a bounded domain with smooth boundary and the nonlinearity ƒ assumes both positive and negative values. Under the assumption that ∈ satisfies certain symmetry conditions we establish two results providing lower bounds on the C⁰(∈^̅) norm of positive solutions. The bounds derived are the same one obtains in dimension n = 1.