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

Abstract

We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^θ\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 $N^{\prime }:=N+\theta >2$. In particular, for $N^{\prime }>2$ and $\tau :=\ell -\theta >-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N^{\prime }+2+2\tau }{N^{\prime }-2}$, while for any $0<p\le p_s$, the only nonnegative solution is $u\equiv 0$.

To appear in Zeitschrift fuer Angewandte Mathematik und Physik (ZAMP)

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