Regularity of the extremal solution for some elliptic problems with advection

Abstract

In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=λf(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$ is a positive nondecreasing convex function, exploding at a finite value $a\in (0, \infty)$. We show that the extremal solution is regular in low dimensional case. In particular, we prove that for the radial case, all extremal solution is regular in dimension two.

18 pages, accepted by Journal of Differential Equations