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Regularity of the extremal solution for some elliptic problems

In this talk, we will 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 articular, we prove that for the radial case, all extremal solutions are regular in dimension two. We recall also some results on the regularity of the extremal solutions for the superlinear case. This is a joint work with X.Luo and D.Ye.