The regularity of a semilinear elliptic system with quadratic growth of gradient

Abstract

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \begin{equation}\label{sys01} Δu=Q(x, u, \nabla u), \end{equation} for $u:{\mathbb{R}}^n\to {\mathbb{R}}^K$, $Q$ has quadratic growth in $\nabla u$. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1,2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for $n=2$) and F. Béthuel (for $n\ge 3$), assert that a $W^{1,n}$ weak solution of harmonic map is always smooth. We extend Béthuel's result to above general system, that a $W^{1,n}$ weak solution of above system is smooth for $n\ge 3$. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a $W^{2,n/2}$ weak solution of such system is always smooth, for $n\ge 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.

15 pages