Positive bounded solutions for semilinear elliptic equations in smooth domains

Abstract

We are concerned with the following semilinear elliptic equation $\Delta u=\lambda f\left( x,u\right) $ in $D,$ subject to some Dirichlet conditions, where $\lambda \geq 0$ is a parameter and $D$ is a smooth domain in $\mathbb{R}^{n}\left( n\geq 3\right) $. Under some appropriate assumptions on the nonnegative nonlinearity term $f\left( x,u\right) ,$ we show the existence of a positive bounded solution for the above semilinear elliptic equation. Our approach is based on Schauder's fixed point Theorem.