A symmetry result for cooperative elliptic systems with singularities

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Stefano Biagi
Università Politecnica della Marche. Dipartimento di Ingegneria Industriale e Scienze Matematiche
Enrico Valdinoci
University of Western Australia. Department of Mathematics and Statistics
Eugenio Vecchi
Università degli Studi di Trento. Dipartimento di Matematica

We obtain symmetry results for solutions of an elliptic system of equationpossessing a cooperative structure. The domain in which the problem is set maypossess "holes" or "small vacancies" (measured in terms of capacity) along which thesolution may diverge.The method of proof relies on the moving plane technique, which needs to besuitably adapted here to take care of the complications arising from the vacancies inthe domain and the analytic structure of the elliptic system.

Paraules clau
elliptic systems, moving plane method, symmetry of solutions

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Biagi, Stefano et al. «A symmetry result for cooperative elliptic systems with singularities». Publicacions Matemàtiques, 2020, vol.VOL 64, núm. 2, p. 621-52, http://raco.cat/index.php/PublicacionsMatematiques/article/view/371205.
Referències
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Amer.Math. Soc. Transl. (2) 21 (1962), 341-354. DOI: 10.1090/trans2/021/09.

A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. PuraAppl. (4) 58 (1962), 303-315. DOI: 10.1007/BF02413056.

E. Berchio, F. Gazzola, and T. Weth, Radial symmetry of positive solutionsto nonlinear polyharmonic Dirichlet problems, J. Reine Angew. Math.2008(620) (2008), 165-183. DOI: 10.1515/CRELLE.2008.052.

H. Berestycki and L. Nirenberg, On the method of moving planes and thesliding method, Bol. Soc. Brasil. Mat. (N.S.) 22(1) (1991), 1-37. DOI: 10.1007/BF01244896[5

S. Biagi, E. Valdinoci, and E. Vecchi, A symmetry result for elliptic systemsin punctured domains, Commun. Pure Appl. Anal. 18(5) (2019), 2819-2833.DOI: 10.3934/cpaa.2019126.

L. Caffarelli, Y. Y. Li, and L. Nirenberg, Some remarks on singular solutionsof nonlinear elliptic equations. II: symmetry and monotonicity via moving planes,in: \Advances in Geometric Analysis", Adv. Lect. Math. (ALM) 21, Int. Press,Somerville, MA, 2012, pp. 97-105.

A. Canino and M. Degiovanni, A variational approach to a class of singularsemilinear elliptic equations, J. Convex Anal. 11(1) (2004), 147-162.[8]

A. Canino, M. Grandinetti, and B. Sciunzi, Symmetry of solutions of somesemilinear elliptic equations with singular nonlinearities, J. Di_erential Equa-tions 255(12) (2013), 4437-4447. DOI: 10.1016/j.jde.2013.08.014.

A. Canino, L. Montoro, and B. Sciunzi, The moving plane method for singularsemilinear elliptic problems, Nonlinear Anal. 156 (2017), 61-69. DOI: 10.1016/j.na.2017.02.009.

A. Canino and B. Sciunzi, A uniqueness result for some singular semilinearelliptic equations, Commun. Contemp. Math. 18(6) (2016), 1550084, 9 pp.DOI: 10.1142/S0219199715500844.

C.-C. Chen and C.-S. Lin, Local behavior of singular positive solutions of semilinearelliptic equations with Sobolev exponent, Duke Math. J. 78(2) (1995),315-334. DOI: 10.1215/S0012-7094-95-07814-4.

F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem,Commun. Contemp. Math. 21(2) (2019), 1850019, 34 pp. DOI: 10.1142/S0219199718500190.

F. Colasuonno and E. Vecchi, Symmetry and rigidity for the hinged compositeplate problem, J. Di_erential Equations 266(8) (2019), 4901-4924. DOI: 10.1016/j.jde.2018.10.011.

M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problemwith a singular nonlinearity, Comm. Partial Di_erential Equations 2(2) (1977),193-222. DOI: 10.1080/03605307708820029.

L. Damascelli and F. Pacella, Symmetry results for cooperative ellipticsystems via linearization, SIAM J. Math. Anal. 45(3) (2013), 1003-1026.DOI: 10.1137/110853534.

D. G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systemsin general domains, NoDEA Nonlinear Di_erential Equations Appl. 1(2)(1994), 119-123. DOI: 10.1007/BF01193947.

F. Esposito, Symmetry and monotonicity properties of singular solutions tosome cooperative semilinear elliptic systems involving critical nonlinearities, Dis-crete Contin. Dyn. Syst. 40(1) (2020), 549-577. DOI: 10.3934/dcds.2020022.


F. Esposito, A. Farina, and B. Sciunzi, Qualitative properties of singular solutionsto semilinear elliptic problems, J. Difierential Equations 265(5) (2018),1962-1983. DOI: 10.1016/j.jde.2018.04.030.

F. Esposito, L. Montoro, and B. Sciunzi, Monotonicity and symmetry ofsingular solutions to quasilinear problems, J. Math. Pures Appl. (9) 126 (2019),214-231. DOI: 10.1016/j.matpur.2018.09.005.

L. C. Evans and R. F. Gariepy, \Measure Theory and Fine Properties ofFunctions", Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton,FL, 2015.

A. Ferrero, F. Gazzola, and T. Weth, Positivity, symmetry and uniquenessfor minimizers of second-order Sobolev inequalities, Ann. Mat. Pura Appl. (4)186(4) (2007), 565-578. DOI: 10.1007/s10231-006-0019-9.

B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties viathe maximum principle, Comm. Math. Phys. 68(3) (1979), 209-243. DOI: 10.1007/BF01221125.

D. Gilbarg and N. S. Trudinger, \Elliptic Partial Di_erential Equations ofSecond Order", Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. DOI: 10.1007/978-3-642-61798-0.

J. Heinonen, T. Kilpelainen, and O. Martio, \Nonlinear Potential Theory ofDegenerate Elliptic Equations", Unabridged republication of the 1993 original,Dover Publications, Inc., Mineola, NY, 2006.

L. Montoro, F. Punzo, and B. Sciunzi, Qualitative properties of singular solutionsto nonlocal problems, Ann. Mat. Pura Appl. (4) 197(3) (2018), 941-964.DOI: 10.1007/s10231-017-0710-z.

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J.Math. 80(4) (1958), 931-954. DOI: 10.2307/2372841.

P. Pucci and J. Serrin, \The Maximum Principle", Progress in NonlinearDi_erential Equations and their Applications 73, Birkhauser Verlag, Basel, 2007.DOI: 10.1007/978-3-7643-8145-5

B. Sciunzi, On the moving plane method for singular solutions to semilinearelliptic equations, J. Math. Pures Appl. (9) 108(1) (2017), 111-123. DOI: 10.1016/j.matpur.2016.10.012.

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal.43 (1971), 304-318. DOI: 10.1007/BF00250468.

B. Sirakov, Some estimates and maximum principles for weakly coupled systemsof elliptic PDE, Nonlinear Anal. 70(8) (2009), 3039-3046. DOI: 10.1016/j.na.2008.12.026.

S. Terracini, On positive entire solutions to a class of equations with a singularcoe_cient and critical exponent, Adv. Di_erential Equations 1(2) (1996),241-264.

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations,J. Di_erential Equations 42(3) (1981), 400-413. DOI: 10.1016/0022-0396(81)90113-3.

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