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

Abstract

We consider positive solutions to the weighted elliptic problem

$$\begin{aligned} -\text{ div } (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\text{ in } {\mathbb {R}}^N \backslash {{\overline{B}}},\quad u=0 \;\; \text{ on } \partial B, \end{aligned}$$

where B is the standard unit ball of ${\mathbb {R}}^N$. We give a complete answer for the existence question for $N':=N+\theta >2$ and $p > 0$. In particular, for $N' > 2$ and $\tau :=\ell -\theta >-2$, it is shown that for $0< p \le p_s:=\frac{N'+2+2 \tau }{N'-2}$, the only nonnegative solution to the problem is $u \equiv 0$. This nonexistence result is new, even for the classical case $\theta = \ell = 0$ and $\frac{N}{N-2} < p \le \frac{N+2}{N-2}$, $N \ge 3$. The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.