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 (0.1) Δ u = Q (x , u , ∇ u) , for u : R n → R K , 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 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 ≥ 3), assert that a W 1 , n weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1) , that a W 1 , 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 W 2 , 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. [ABSTRACT FROM AUTHOR]