Positive solutions of $\Delta u+K(x)u^ p=0$ without decay conditions on $K(x)$

Abstract:

This paper deals with the existence of positive solutions of the nonlinear elliptic equation $\Delta u + K(x){u^p} = 0$ in ${R^n}$ with $n \geq 3$ and $\tfrac {n}{{n - 2}} < p < \tfrac {{n + 2}}{{n - 2}}$, where $K(x)$ does not decay at $\infty$. The existence of classical positive solutions and singular positive solutions is proved under the hypothesis that $K$ is radial symmetric, $K(r) = 1 + H(r)$ is a perturbation of the constant 1, and $H(r)$ satisfies some conditions.