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A Berestycki–Lions type result for a class of degenerate elliptic problems involving the Grushin operator

Part of: Elliptic equations and systems Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  05 July 2022

Claudianor O. Alves  and
Angelo R. F. de Holanda
Claudianor O. Alves
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, 58429-970, Campina Grande, PB, Brazil (coalves@mat.ufcg.edu.br, angelo@mat.ufcg.edu.br)
Angelo R. F. de Holanda
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, 58429-970, Campina Grande, PB, Brazil (coalves@mat.ufcg.edu.br, angelo@mat.ufcg.edu.br)
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Abstract

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations

\[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \]
where $\Delta _{\gamma }$ is known as the Grushin operator, $z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$ and $m+k=N\geqslant 3$, $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$, we have showed a Berestycki–Lions type result.

Keywords

Variational methodsdegenerate elliptic equationsP.D.E. on manifolds

MSC classification

Primary: 35A15: Variational methods
Secondary: 35J70: Degenerate elliptic equations 35R01: Partial differential equations on manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1244 - 1271
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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