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

TL;DR

This paper investigates the regularity of solutions to semilinear elliptic systems with quadratic gradient growth, extending classical harmonic map results to higher dimensions and orders, and demonstrating optimal regularity conditions.

Contribution

It extends regularity results for harmonic and biharmonic map systems to more general elliptic systems with quadratic growth, establishing optimal smoothness conditions for weak solutions.

Findings

Weak solutions are smooth for n≥3 in second-order systems.

Weak solutions are smooth for n≥5 in fourth-order systems.

Constructed examples show the optimality of regularity conditions.

Abstract

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.