On the existence of a positive solution for a semilinear elliptic equation in the upper half space

Abstract

We show the existence of a nontrivial solution to the semilinear elliptic equation $-\Delta u+u=b\left(x\right){\left|u\right|}^{p-2}u,u>0,u\in H_0^1\left({ℝ}_{+}^N\right)$ under some suitable conditions.