Electron. J. Diff. Equ., Vol. 2015 (2015), No. 86, pp. 1-15.

Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth

Giovany M. Figueiredo, Marcos T. O. Pimenta

Abstract:
In this work we study an existence and multiplicity of solutions for the prescribed mean-curvature problem with critical growth,
$$\displaylines{ -\hbox{div}\Bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigl) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u \quad \text{in }\Omega \cr u = 0 \quad \text{on } \partial \Omega, }$$
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ and $1 <q<2$. To employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions of the modified problem is necessary to recover solutions of the original problem.

Submitted March 24, 2014. Published April 7, 2015.
Math Subject Classifications: 35J93, 35J62, 35J20.
Key Words: Prescribed mean-curvature problem; critical exponent; variational methods.

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Giovany M. Figueiredo
Universidade Federal do Pará
Faculdade de Matemática
CEP: 66075-110 Belém - Pa, Brazil
email: giovany@ufpa.br
Marcos T. O. Pimenta
Faculdade de Ciências e Tecnlogia
UNESP - Univ Estadual Paulista
19060-900, Presidente Prudente - SP, Brazil
Phone (55) 18 - 3229-5625 FAX (55) 18 - 3221-8333
email: pimenta@fct.unesp.br

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