Geodesic
Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth
Electronic journal of differential equations, Tome 2015 (2015)
In this work we study an existence and multiplicity of solutions for the prescribed mean-curvature problem with critical growth,

$\displaylines{ -\hbox{div}\Bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigl) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u \quad {in }\Omega \cr u = 0 \quad {on } \partial \Omega, }$

where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}, N\geq 3$ and $1 $. 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.
Classification : 35J93, 35J62, 35J20
Keywords: prescribed mean-curvature problem, critical exponent, variational methods 
@article{EJDE_2015__2015__a87,
     author = {Figueiredo,  Giovany M. and Pimenta,  Marcos T.O.},
     title = {Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth},
     journal = {Electronic journal of differential equations},
     year = {2015},
     volume = {2015},
     zbl = {1315.35116},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a87/}
}
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%F EJDE_2015__2015__a87
Figueiredo,  Giovany M.; Pimenta,  Marcos T.O. Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth. Electronic journal of differential equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a87/