# Existence and regularity of extremal solutions for a mean-curvature equation

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1 Department of Mathematics, University of Maryland 2 ICJ - Institut Camille Jordan Villeurbanne

Abstract : We study a class of semi-linear mean curvature equations $\mathcal Mu=H+\lambda f u$ where $\mathcal M$ is the mean curvature operator. We show that there exists an extremal parameter $\lambda^*$ such that this equation admits a minimal weak solutions for all $\lambda \in 0,\lambda^*$, while no weak solutions exists for $\lambda >\lambda^*$. In the radial case, we then show that minimal solutions are classical solutions for all $\lambda\in 0,\lambda^*$ and that another branch of solution exists in a neighborhood $\lambda *-\eta,\lambda^*$ of $\lambda^*$.

Keywords : mean curvature operator semilinear equation minimal solutions regularity

Author: ** Antoine Mellet - Julien Vovelle - **

Source: https://hal.archives-ouvertes.fr/