Nonexistence of positive solutions for Henon equation

We consider the semilinear elliptic equation $$-\Delta u = |x|^\alpha u^p \quad \hbox{in } \mathbb{R}^N,$$ where $N\ge 3$, $\alpha>-2$ and $p>1$. We show that there are no positive solutions provided that the exponent $p$ additionally verifies $$1<p<\frac{N+2\alpha+2}{N-2}.$$ 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 $-2<\alpha<0$.