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

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \begin{equation}\label{sys01} \Delta u=Q(x, u, \nabla u), \end{equation} for $u: \mathbb{R}^n\rightarrow \mathbb{R}^K$, $Q$ 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. H\'elein (for $n=2$) and F. B\'ethuel (for $n\geq 3$), assert that a $W^{1, n}$ weak solution of harmonic map is always smooth. We extend B\'ethuel's result to above general system, that a $W^{1, n}$ weak solution of above system is smooth for $n\geq 3$. 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 $n\geq 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.