A symmetry result for cooperative elliptic systems with singularities. (English) Zbl 1447.35148

The authors investigate symmetry results for the solutions of an elliptic system of equations possessing a cooperative structure in a domain that may have “holes” or “cuts” or, more in general, “small vacancies” along which the solutions can become singular. More precisely, they consider a convex open set \(\Omega \subseteq \mathbb{R}^n\) of class \(C^\infty\) which is bounded and symmetric with respect to the hyperplane \(\{x_1=0\}\) and a closed set \(\Gamma \subseteq \Omega \cap \{x_1=0\}\) which consists of a point if \(n=2\) or verifying \(\mathrm{Cap}_2(\Gamma)=0\) if \(n\geq 3\). Then they analyse a second-order cooperative elliptic system \[ \left\{ \begin{array}{ll} -\Delta u_i =f_i(u_1,\dots,u_m) & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i>0 & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i \equiv 0 & \mathrm{on}\ \partial \Omega\, , \end{array} \right. \] where \(f_1,\dots, f_m \in \mathrm{Lip}(\mathbb{R}^m)\). Under suitable assumptions, they show that the solutions \(u_1, \dots u_m\) are symmetric with respect to the hyperplane \(\{x_1=0\}\) and increasing in the \(x_1\)-direction in \(\Omega \cap \{x_1<0\}\) and that for every \(i \in {1,\dots,m}\) one has \[ \frac{\partial u_i}{\partial x_1}(x) \] for every \(x \in \Omega \cap \{x_1<0\}\).


35J57 Boundary value problems for second-order elliptic systems
35B06 Symmetries, invariants, etc. in context of PDEs
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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[1] 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. · Zbl 0122.39601
[2] A. D. Alexandrov, A characteristic property of spheres,Ann. Mat. Pura Appl. (4)58(1962), 303-315.DOI: 10.1007/BF02413056. · Zbl 0107.15603
[3] E. Berchio, F. Gazzola, and T. Weth, Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems,J. Reine Angew. Math. 2008(620)(2008), 165-183.DOI: 10.1515/CRELLE.2008.052. · Zbl 1182.35109
[4] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method,Bol. Soc. Brasil. Mat. (N.S.)22(1)(1991), 1-37.DOI: 10.1007/ BF01244896 · Zbl 0784.35025
[5] S. Biagi, E. Valdinoci, and E. Vecchi, A symmetry result for elliptic systems in punctured domains,Commun. Pure Appl. Anal.18(5)(2019), 2819-2833. DOI: 10.3934/cpaa.2019126.
[6] L. Caffarelli, Y. Y. Li, and L. Nirenberg, Some remarks on singular solutions of 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. · Zbl 1325.35044
[7] A. Canino and M. Degiovanni, A variational approach to a class of singular semilinear elliptic equations,J. Convex Anal.11(1)(2004), 147-162. · Zbl 1073.35092
[8] A. Canino, M. Grandinetti, and B. Sciunzi, Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities,J. Differential Equations255(12)(2013), 4437-4447.DOI: 10.1016/j.jde.2013.08.014. · Zbl 1286.35011
[9] A. Canino, L. Montoro, and B. Sciunzi, The moving plane method for singular semilinear elliptic problems,Nonlinear Anal.156(2017), 61-69.DOI: 10. 1016/j.na.2017.02.009. · Zbl 1378.35132
[10] A. Canino and B. Sciunzi, A uniqueness result for some singular semilinear elliptic equations,Commun. Contemp. Math.18(6)(2016), 1550084, 9 pp. DOI: 10.1142/S0219199715500844. · Zbl 1458.35179
[11] C.-C. Chen and C.-S. Lin, Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent,Duke Math. J.78(2)(1995), 315-334.DOI: 10.1215/S0012-7094-95-07814-4. · Zbl 0839.35014
[12] F. Colasuonno and E. Vecchi, Symmetry in the composite plate problem,Commun. Contemp. Math.21(2)(2019), 1850019, 34 pp.DOI: 10.1142/ S0219199718500190. · Zbl 1416.35102
[13] F. Colasuonno and E. Vecchi, Symmetry and rigidity for the hinged composite plate problem,J. Differential Equations266(8)(2019), 4901-4924.DOI: 10. 1016/j.jde.2018.10.011. · Zbl 1423.31003
[14] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity,Comm. Partial Differential Equations2(2)(1977), 193-222.DOI: 10.1080/03605307708820029. · Zbl 0362.35031
[15] L. Damascelli and F. Pacella, Symmetry results for cooperative elliptic systems via linearization,SIAM J. Math. Anal.45(3)(2013), 1003-1026. DOI: 10.1137/110853534. · Zbl 1284.35166
[16] D. G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains,NoDEA Nonlinear Differential Equations Appl.1(2) (1994), 119-123.DOI: 10.1007/BF01193947. · Zbl 0822.35039
[17] F. Esposito, Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities,Discrete Contin. Dyn. Syst.40(1)(2020), 549-577.DOI: 10.3934/dcds.2020022. · Zbl 1433.35055
[18] F. Esposito, A. Farina, and B. Sciunzi, Qualitative properties of singular solutions to semilinear elliptic problems,J. Differential Equations265(5)(2018), 1962-1983.DOI: 10.1016/j.jde.2018.04.030. · Zbl 1394.35188
[19] F. Esposito, L. Montoro, and B. Sciunzi, Monotonicity and symmetry of singular solutions to quasilinear problems,J. Math. Pures Appl. (9)126(2019), 214-231.DOI: 10.1016/j.matpur.2018.09.005. · Zbl 1418.35199
[20] L. C. Evans and R. F. Gariepy,“Measure Theory and Fine Properties of Functions”, Revised edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015. · Zbl 1310.28001
[21] A. Ferrero, F. Gazzola, and T. Weth, Positivity, symmetry and uniqueness for minimizers of second-order Sobolev inequalities,Ann. Mat. Pura Appl. (4) 186(4)(2007), 565-578.DOI: 10.1007/s10231-006-0019-9. · Zbl 1141.46014
[22] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle,Comm. Math. Phys.68(3)(1979), 209-243.DOI: 10. 1007/BF01221125. · Zbl 0425.35020
[23] D. Gilbarg and N. S. Trudinger,“Elliptic Partial Differential Equations of Second Order”, Reprint of the 1998 edition, Classics in Mathematics, SpringerVerlag, Berlin, 2001.DOI: 10.1007/978-3-642-61798-0. · Zbl 1042.35002
[24] J. Heinonen, T. Kilpel¨ainen, and O. Martio,“Nonlinear Potential Theory of Degenerate Elliptic Equations”, Unabridged republication of the 1993 original, Dover Publications, Inc., Mineola, NY, 2006.
[25] L. Montoro, F. Punzo, and B. Sciunzi, Qualitative properties of singular solutions to nonlocal problems,Ann. Mat. Pura Appl. (4)197(3)(2018), 941-964. DOI: 10.1007/s10231-017-0710-z. · Zbl 1391.35384
[26] J. Nash, Continuity of solutions of parabolic and elliptic equations,Amer. J. Math.80(4)(1958), 931-954.DOI: 10.2307/2372841. · Zbl 0096.06902
[27] P. Pucci and J. Serrin,“The Maximum Principle”, Progress in Nonlinear Differential Equations and their Applications73, Birkh¨auser Verlag, Basel, 2007. DOI: 10.1007/978-3-7643-8145-5.
[28] B. Sciunzi, On the moving plane method for singular solutions to semilinear elliptic equations,J. Math. Pures Appl. (9)108(1)(2017), 111-123.DOI: 10. 1016/j.matpur.2016.10.012. · Zbl 1371.35114
[29] J. Serrin, A symmetry problem in potential theory,Arch. Rational Mech. Anal. 43(1971), 304-318.DOI: 10.1007/BF00250468. · Zbl 0222.31007
[30] B. Sirakov, Some estimates and maximum principles for weakly coupled systems of elliptic PDE,Nonlinear Anal.70(8)(2009), 3039-3046.DOI: 10.1016/j.na. 2008.12.026. · Zbl 1173.35391
[31] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent,Adv. Differential Equations1(2)(1996), 241-264. · Zbl 0847.35045
[32] W.
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