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arXiv · 1810.07343

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

Zongming Guo·Xia Huang·Dong Ye
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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 $N':=N+θ>2$. In particular, for $N' > 2$ and $τ:=\ell-θ>-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2τ}{N'-2}$, while for any $ 0<p \leq p_s$, the only nonnegative solution is $u \equiv 0$.

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Zongming Guo, Xia Huang, Dong Ye. 2020-06-08. Existence and nonexistence results for a weighted elliptic equation in exterior domains. https://doi.org/10.1007/s00033-020-01338-0

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