Regularity of minimizers of semilinear elliptic problems up to dimension four

Details Regularity of minimizers of semilinear elliptic problems up to dimension four Regularity of minimizers of semilinear elliptic problems up to dimension four

2009-09-25

arxiv.org/abs/0909.4696v2

Mathematics

Analysis of PDEs

Scientific paper

We consider the class of semi-stable solutions to semilinear equations $-\Delta u=f(u)$ in a bounded smooth domain $\Omega$ of $R^n$ (with $\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n \leq 4$, we establish an priori $L^\infty$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $\Omega$ is convex. This result was previously known only in dimensions $n\leq 3$ by a result of G. Nedev. In dimensions $5 \leq n \leq 9$ the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case $\Omega=B_R$ by a result of A. Capella and the author.

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