Nonexistence of positive solutions for Henon equation

We consider the semilinear elliptic equation $$-\mathrm{\Delta }u=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{u}^p\text{in }{\mathbb{R}}^N,$$ where $N\ge 3$, $\alpha >-2$ and $p>1$. We show that there are no positive solutions provided that the exponent $p$ additionally verifies $$1<p<\frac{N+2\alpha +2}{N-2}\mathrm{.}$$ This solves an open problem posed in previous literature, where only the radially symmetric case was fully understood. We also characterize all positive solutions when $p=\frac{N+2\alpha +2}{N-2}$ and $-2<\alpha <0$.