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.

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Acknowledgements

Z.G is supported by NSFC (No. 11571093 and 11171092). X.H is supported by NSFC (No. 11701181). D.Y is partially supported by Science and Technology Commission of Shanghai Municipality (STCSM) (No. 18dz2271000). The authors thank the anonymous referees for their careful reading.

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Correspondence to Dong Ye.

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Guo, Z., Huang, X. & Ye, D. Existence and nonexistence results for a weighted elliptic equation in exterior domains. Z. Angew. Math. Phys. 71, 116 (2020). https://doi.org/10.1007/s00033-020-01338-0

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Keywords

  • Weighted elliptic equation
  • Existence and nonexistence
  • Critical exponent
  • Exterior domain

Mathematics Subject Classification

  • 35B09
  • 35J60
  • 35J25
  • 35R37