Semilinear elliptic problems involving a fast increasing diffusion weight

Abstract

In this work we study the existence and multiplicity of positive bounded solutions to a class of problems with a reaction–diffusion equation: $\left(\begin{array}{ccc}-\text{div}\left(K\left(x\right)\nabla u\right)=f(x,u)& \text{in}& R^N,\\ u\left(x\right)\to 0& \text{as}& \left|x\right|\to \infty .\end{array}\right)$Using a sublinear hypothesis on the nonlinearity $f(x,u)$ near the origin, we obtain a solution $u_1.$ Furthermore taking $K\left(x\right)=e^{\theta \left(x\right)}$ where $\theta \left(x\right)$ satisfies some fast increasing growth conditions, we find via variational methods, a second solution $u_2$ in such a way that $u_2>u_1.$ For this purpose, a new type of compactness is provided for the associated energy functional.