Regularity of the extremal solution for some elliptic problems with advection

TL;DR

This paper studies the regularity of extremal solutions to certain semilinear elliptic equations with advection, demonstrating regularity in low dimensions, especially in the radial case for dimension two.

Contribution

It establishes the regularity of extremal solutions for elliptic problems with advection in low dimensions, including the radial case in two dimensions.

Findings

Extremal solutions are regular in low dimensions.

Radial extremal solutions are regular in dimension two.

Regularity results depend on the dimension and symmetry of the problem.

Abstract

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