Nonexistence of positive solutions for Henon equation

Abstract

We consider the semilinear elliptic equation $$-\Delta u=|x{|}^{\alpha }{u}^{p}in{\mathbb{R}}^N,$$ 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 $-2<\alpha <0$.

References in corpus (1)

• Nonradial solutions for the Hénon equation in $R^N$