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Existence and multiplicity of positive solutions for a fourth-order elliptic equation

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  28 January 2019

Giovany M. Figueiredo ,
Marcelo F. Furtado  and
João Pablo P. da Silva
Giovany M. Figueiredo
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasilia-DF, Brazil ([email protected]; [email protected])
Marcelo F. Furtado
Affiliation:
Universidade de Brasília, Departamento de Matemática, 70910-900 Brasilia-DF, Brazil ([email protected]; [email protected])
João Pablo P. da Silva
Affiliation:
Universidade Federal do Pará, Departamento de Matemática, 66075-100 Belém-PA, Brazil ([email protected])
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Abstract

We prove existence and multiplicity of solutions for the problem

$$\left\{ {\matrix{ {\Delta ^2u + \lambda \Delta u = \vert u \vert ^{2*-2u},{\rm in }\Omega ,} \hfill \hfill \hfill \hfill \cr {u,-\Delta u > 0,\quad {\rm in}\;\Omega ,\quad u = \Delta u = 0,\quad {\rm on}\;\partial \Omega ,} \cr } } \right.$$
where $\Omega \subset {\open R}^N$, $N \ges 5$, is a bounded regular domain, $\lambda >0$ and $2^*=2N/(N-4)$ is the critical Sobolev exponent for the embedding of $W^{2,2}(\Omega )$ into the Lebesgue spaces.

Keywords

Fourth-order problemscritical nonlinearitiesvariational methods

MSC classification

Primary: 35J35: Variational methods for higher-order elliptic equations
Secondary: 35J30: Higher-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 1053 - 1069
Copyright
Copyright © Royal Society of Edinburgh 2019

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