Regularity of the extremal solution for some elliptic problems with advection

In this note, we investigate the regularity of extremal solution $u^{\ast }$ for semilinear elliptic equation $-\mathrm{△}u+c(x)\cdot \mathrm{\nabla }u=\lambda f(u)$ 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 $a\in (0,\mathrm{\infty })$. 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.