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

Xue Luo (1) , Dong Ye (2) , Feng Zhou (1)

1 ECNU - East China Normal University [Shangaï]
2 LMAM - Laboratoire de Mathématiques et Applications de Metz

In this note, we investigate the regularity of the extremal solution u * for the semilinear elliptic equation − u + c(x) • ∇u = λf (u) on a bounded smooth domain of R 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.

Mots clés: regularity; extremal solution; advection; singular nonlinearity

Domaines: Equations aux dérivées partielles [math.AP]

Xue Luo, Dong Ye, Feng Zhou. Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection. Journal of Differential Equations, 2011, 251 (8), pp.2082-2099. ⟨10.1016/j.jde.2011.07.011⟩. ⟨hal-03303085⟩