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(0.1)Δu=Q(x,u,∇u), for u:Rn→RK, Q has quadratic growth in ∇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 W1,2 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≥3), assert that a W1,n weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1), that a W1,n weak solution of the system is smooth for n≥3. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a W2,n/2 weak solution of such system is always smooth, for n≥5. We also construct various examples, and these examples show that our regularity results are optimal in various sense.