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

Abstract

We will consider the nonlocal H\'enon equation $$(-\Delta)^s u= |x|^{\alpha} u^{p},\quad \mathbb{R}^{N},$$ 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 $$1<p<p^*_{\alpha, s}:=\frac{N+2\alpha+2s}{N-2s}.$$ Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solutions, exists when $p=p_{\alpha, s}^{*}$.