Semilinear Elliptic Problems Involving a Fast Increasing Diffusion Weight

In this work we study the existence and multiplicity of positive bounded solutions to a class of problems with a reaction–diffusion equation: −divK(x)∇u=f(x,u)inRN,u(x)→0as|x|→∞.Using a sublinear hypothesis on the nonlinearity f(x,u) near the origin, we obtain a solution u1. Furthermore taking K(x)=eθ(x) where θ(x) satisfies some fast increasing growth conditions, we find via variational methods, a second solution u2 in such a way that u2>u1. For this purpose, a new type of compactness is provided for the associated energy functional.