Existence and regularity of extremal solutions for a mean-curvature equation
Résumé
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 $\lambda^*$.
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