Existence and nonexistence results for a weighted elliptic equation in exterior domains

TL;DR

This paper investigates the existence and uniqueness of positive solutions to a weighted elliptic equation in exterior domains, establishing precise conditions on parameters for when solutions exist or are trivial.

Contribution

It provides a complete characterization of solution existence for a weighted elliptic problem in exterior domains, including a unique radial solution for supercritical exponents.

Findings

Unique positive radial solution for p > p_s

Nonexistence of nonnegative solutions for 0 < p ≤ p_s

Explicit critical exponent p_s depending on parameters

Abstract

We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+\theta>2$. In particular, for $N' > 2$ and $\tau:=\ell-\theta >-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2\tau}{N'-2}$, while for any $ 0<p \leq p_s$, the only nonnegative solution is $u \equiv 0$.