Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

Abstract

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problem $\Delta u=\lambda p\left(x\right)f\left(u\right)$ in $\Omega$ subject to some Dirichlet conditions, where $\Omega$ is a regular domain in ${ℝ}^n$ $\left(n\ge 3\right)$ with compact boundary. The nonlinearity $f$ is nonnegative continuous and the potential $p$ belongs to some Kato class $K\left(\Omega \right)$. So we prove the existence of a positive continuous solution depending on $\lambda$ by the use of a potential theory approach.