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

Abstract : We consider positive solutions to the weighted elliptic problem \begin{equation*} -\mbox{div} (|x|^\theta \nabla u)=|x|^\ell u^p \;\;\mbox{in $\mathbb{R}^N \backslash {\overline B}$},\quad u=0 \;\; \mbox{on $\partial B$}, \end{equation*} 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 $0< p \leq p_s:=\frac{N'+2+2 \tau}{N'-2}$, 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 $\frac{N}{N-2} < p \leq \frac{N+2}{N-2}$, $N \geq 3$. The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption.
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https://hal.univ-lorraine.fr/hal-02894451
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Submitted on : Wednesday, July 8, 2020 - 10:50:13 PM
Last modification on : Monday, July 20, 2020 - 11:19:38 AM
Long-term archiving on: : Monday, November 30, 2020 - 4:46:19 PM

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Zongming Guo, Xia Huang, Dong Ye. Existence and nonexistence results for a weighted elliptic equation in exterior domains. Zeitschrift für Angewandte Mathematik und Physik, Springer Verlag, 2020, 71 (4), ⟨10.1007/s00033-020-01338-0⟩. ⟨hal-02894451⟩

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