The sharp exponent in the study of the nonlocal Hénon equation in ${\mathbb{R}}^n$. A Liouville theorem and an existence result

Abstract

We will consider the nonlocal Hénon equation $$(-\Delta {)}^{s}u=|x{|}^{\alpha }{u}^p,{\mathbb{R}}^N,$$ where $(-\Delta {)}^s$ is the fractional Laplacian operator with $0<s<1$, $-2s<\alpha$, $p>1$ and $N>2s$. We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when $$1<p<p_{\alpha ,s}^{\ast }:=\frac{N+2\alpha +2s}{N-2s}.$$ Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solutions, exists when $p=p_{\alpha ,s}^{\ast }$.