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

TL;DR

This paper investigates the nonlocal Hénon equation involving the fractional Laplacian, establishing a Liouville theorem for nonexistence of solutions below a critical exponent and proving the existence of bubble solutions at the critical exponent.

Contribution

It provides the first nonexistence result in the optimal range and demonstrates the existence of bubble solutions at the critical exponent for the nonlocal Hénon equation.

Findings

Nonexistence of positive solutions for subcritical exponents.

Existence of bubble solutions at the critical exponent.

Identification of the sharp exponent in the nonlocal Hénon equation.

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 $p=p_{\alpha, s}^{*}$.