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 u<inf>1</inf>. Furthermore taking K(x)=eθ(x) where θ(x) satisfies some fast increasing growth conditions, we find via variational methods, a second solution u<inf>2</inf> in such a way that u<inf>2</inf>>u<inf>1</inf>. For this purpose, a new type of compactness is provided for the associated energy functional. © 2024 Elsevier Ltd