June  2014, 34(6): 2657-2667. doi: 10.3934/dcds.2014.34.2657

On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Received  February 2013 Published  December 2013

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain $\Omega$. We assume that $\Omega$ is symmetric about a hyperplane $H$ and convex in the direction perpendicular to $H$. Each nonnegative solution of such a problem is symmetric about $H$ and, if strictly positive, it is also decreasing in the direction orthogonal to $H$ on each side of $H$. The latter is of course not true if the solution has a nontrivial nodal set. In this paper we prove that for a class of domains, including for example all domains which are convex (in all directions), there can be at most one nonnegative solution with a nontrivial nodal set. For general domains, there are at most finitely many such solutions.
Citation: Peter Poláčik. On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2657-2667. doi: 10.3934/dcds.2014.34.2657
References:
[1]

H. Berestycki, Qualitative properties of positive solutions of elliptic equations, in Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000, 34-44.

[2]

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

[3]

X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.

[4]

A. Castro and R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations, 14 (1989), 1091-1100. doi: 10.1080/03605308908820645.

[5]

X.-Y. Chen, On the scaling limits at zeros of solutions of parabolic equations, J. Differential Equations, 147 (1998), 355-382. doi: 10.1006/jdeq.1997.3329.

[6]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. European Math. Soc., 9 (2007), 317-330. doi: 10.4171/JEMS/81.

[7]

L. Damascelli, F. Pacella and M. Ramaswamy, A strong maximum principle for a class of non-positone singular elliptic problems, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 187-196.

[8]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications, Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.

[9]

J. Földes, On symmetry properties of parabolic equations in bounded domains, J. Differential Equations, 250 (2011), 4236-4261. doi: 10.1016/j.jde.2011.03.018.

[10]

, J. Földes and P. Poláčik,, Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains, (). 

[11]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, 128, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.

[12]

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

[13]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[14]

P. Hess and P. Poláčik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 573-587. doi: 10.1017/S030821050002878X.

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.

[16]

B. Kawohl, Symmetrization - or how to prove symmetry of solutions to a PDE, in Partial Differential Equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000, 214-229.

[17]

F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308. doi: 10.1002/cpa.3160440303.

[18]

C. Miranda, Partial Differential Equations of Elliptic Type, Second revised edition, Translated from the Italian by Zane C. Motteler, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970.

[19]

W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary Partial Differential Equations, Vol. 1 (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6.

[20]

P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (eds. W.-Y. Lin Y. Du and H. Ishii), World Scientific, Hackensack, NJ, 2009, 170-208. doi: 10.1142/9789812834744_0009.

[21]

________, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations, 36 (2011), 657-669. doi: 10.1080/03605302.2010.513026.

[22]

________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 29 (2012), 1-19. doi: 10.1016/j.anihpc.2011.03.001.

[23]

________, Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains, J. Funct. Anal., 262 (2012), 4458-4474. doi: 10.1016/j.jfa.2012.02.022.

[24]

P. Poláčik, A discussion of nonnegative solutions of elliptic equations on symmetric domains, to appear in Proceedings of the RIMS Conference on Partial Differential Equations, 2013.

[25]

P. Poláčik and S. Terracini, Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains,, to appear in Proc. AMS., (). 

[26]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007.

show all references

References:
[1]

H. Berestycki, Qualitative properties of positive solutions of elliptic equations, in Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000, 34-44.

[2]

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

[3]

X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570. doi: 10.1002/cpa.3160480504.

[4]

A. Castro and R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations, 14 (1989), 1091-1100. doi: 10.1080/03605308908820645.

[5]

X.-Y. Chen, On the scaling limits at zeros of solutions of parabolic equations, J. Differential Equations, 147 (1998), 355-382. doi: 10.1006/jdeq.1997.3329.

[6]

F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. European Math. Soc., 9 (2007), 317-330. doi: 10.4171/JEMS/81.

[7]

L. Damascelli, F. Pacella and M. Ramaswamy, A strong maximum principle for a class of non-positone singular elliptic problems, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 187-196.

[8]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol. 1. Maximum Principles and Applications, Series in Partial Differential Equations and Applications, 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.

[9]

J. Földes, On symmetry properties of parabolic equations in bounded domains, J. Differential Equations, 250 (2011), 4236-4261. doi: 10.1016/j.jde.2011.03.018.

[10]

, J. Földes and P. Poláčik,, Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains, (). 

[11]

L. E. Fraenkel, An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Mathematics, 128, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511569203.

[12]

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

[13]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[14]

P. Hess and P. Poláčik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 573-587. doi: 10.1017/S030821050002878X.

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Reprint of the 1994 edition, Classics in Mathematics, Springer, Berlin, 2007.

[16]

B. Kawohl, Symmetrization - or how to prove symmetry of solutions to a PDE, in Partial Differential Equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, 2000, 214-229.

[17]

F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308. doi: 10.1002/cpa.3160440303.

[18]

C. Miranda, Partial Differential Equations of Elliptic Type, Second revised edition, Translated from the Italian by Zane C. Motteler, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970.

[19]

W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary Partial Differential Equations, Vol. 1 (eds. M. Chipot and P. Quittner), Handb. Differ. Equ., North-Holland, Amsterdam, 2004, 157-233. doi: 10.1016/S1874-5733(04)80005-6.

[20]

P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions (eds. W.-Y. Lin Y. Du and H. Ishii), World Scientific, Hackensack, NJ, 2009, 170-208. doi: 10.1142/9789812834744_0009.

[21]

________, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations, 36 (2011), 657-669. doi: 10.1080/03605302.2010.513026.

[22]

________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 29 (2012), 1-19. doi: 10.1016/j.anihpc.2011.03.001.

[23]

________, Positivity and symmetry of nonnegative solutions of semilinear elliptic equations on planar domains, J. Funct. Anal., 262 (2012), 4458-4474. doi: 10.1016/j.jfa.2012.02.022.

[24]

P. Poláčik, A discussion of nonnegative solutions of elliptic equations on symmetric domains, to appear in Proceedings of the RIMS Conference on Partial Differential Equations, 2013.

[25]

P. Poláčik and S. Terracini, Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains,, to appear in Proc. AMS., (). 

[26]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73, Birkhäuser Verlag, Basel, 2007.

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