Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u* for the semilinear elliptic equation -Δu+c(x)·u=Λf(u) on a bounded smooth domain of R{double-struck}ⁿ with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a⊂(0,∞). We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.