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

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

2009-04-03

arxiv.org/abs/0904.0618v2

Mathematics

Analysis of PDEs

v2: Typos corrected. Proof of Theorem 2.11 added.

Scientific paper

We study a class of mean curvature equations $-\mathcal Mu=H+\lambda u^p$ where $\mathcal M$ denotes the mean curvature operator and for $p\geq 1$. 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 $\lambda >\lambda^*$ (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all $\lambda\in [0,\lambda^*]$ and that another branch of classical solutions exists in a neighborhood $(\lambda_*-\eta,\lambda^*)$ of $\lambda^*$.

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