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 fo...
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Published in | Journal of functional analysis Vol. 276; no. 4; pp. 1294 - 1312 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
15.02.2019
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0022-1236 1096-0783 |
DOI: | 10.1016/j.jfa.2018.10.007 |