Symmetry of solutions of semilinear elliptic problems

Journal of the European Mathematical Society

Volumn 10, Issue 2, 2008, Pages 439-456

  Van Schaftingen, Jean ^a   Willem, Michel ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Foliated Schwarz symmetrization; Intermediate value theorem; Partial symmetry of solutions to semilinear elliptic equations; Polarization; Quasi continuous functions; Spherical cap symmetrization; Steiner symmetrization; Symmetrization

Indexed keywords

EID: 43649097607     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. Ancona, A.: Une propriété d'invariance des ensembles absorbants par perturbation d'un opérateur elliptique. Comm. Partial Differential Equations 4, 321-337 (1979) Zbl 0459.35027 MR 0525774
2. Bartsch, T.: Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117-152 (2001) Zbl pre01679364 MR 1863294
3. Bartsch, T., Weth, T.: A note on additional properties of sign changing solutions to super-linear elliptic equations. Topol. Methods Nonlinear Anal. 22, 1-14 (2003) Zbl pre02096714 MR 2037264
4. Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1-18 (2005) Zbl pre05010201 MR 2177179
5. Brezis, H., Ponce, A. C.: Remarks on the strong maximum principle. Differential Integral Equations 16, 1-12 (2003) Zbl pre02004993 MR 1948870
6. Brezis, H., Ponce, A. C.: Kato's inequality when Au is a measure. C. R. Math. Acad. Sci. Paris 338, 599-604 (2004) Zbl pre02093937 MR 2056467
7. Brock, F., Solynin, A. Yu.: An approach to symmetrization via polarization. Trans. Amer. Math. Soc. 352, 1759-1796 (2000) Zbl 0965.49001 MR 1695019
8. Castro, A., Drábek, P., Neuberger, J. M.: A sign-changing solution for a superlinear Dirichlet problem. II. In: Proc. Fifth Mississippi State Conf. on Differential Equations and Computational Simulations, Electron. J. Differ. Equ. Conf. 10, Southwest Texas State Univ., 101-107 (2003) Zbl pre01911317 MR 1976635
9. Chabi, A., Haraux, A.: Un théorème de valeurs intermédiaires dans les espaces de Sobolev et applications. Ann. Fac Sci. Toulouse Math. (5) 7, 87-100 (1985) Zbl 0557.35055 MR 0842764
10. Chern, J.-L., Lin, C.-S.: The symmetry of least-energy solutions for semilinear elliptic equations. J. Differential Equations 187, 240-268 (2003) Zbl pre01884221 MR 1949440
11. Evans, L. C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992) Zbl 0804.28001 MR 1158660
12. Lieb, E. H., Loss, M.: Analysis. 2nd ed., Grad. Stud. Math. 14, Amer. Math. Soc., Providence, RI (2001) Zbl 0966.26002 MR 1817225
13. Lin, C.-S., Takagi, I.: Method of rotating planes applied to a singularly perturbed Neumann problem. Calc Var. Partial Differential Equations 13, 519-536 (2001) Zbl pre01713845 MR 1867940
14. Liu, Z., Wang, Z.-Q.: On the Ambrosetti-Rabinowitz superlinear condition. Adv. Nonlinear Stud. 4, 563-574 (2004) Zbl pre02149279 MR 2100913
15. Smets, D., Willem, M.: Partial symmetry and asymptotic behavior for some elliptic problems. Calc Var. Partial Differential Equations 18, 57-75 (2003) Zbl pre01993491 MR 2001882
16. Van Schaftingen, J.: Approximation of symmetrizations and symmetry of critical points. Topol. Methods Nonlinear Anal. 28, 61-85 (2006) Zbl 1110.28014 MR 2262256
17. Van Schaftingen, J.: Universal approximation of symmetrizations by polarizations. Proc. Amer. Math. Soc. 134, 177-186 (2006) Zbl pre02212306 MR 2170557
18. Van Schaftingen, J., Willem, M.: Set transformations, symmetrizations and isoperimetric in-equalities. In: Nonlinear Analysis and Applications to the Physical Sciences, V. Benci and A. Masiello (eds.), Springer, 135-152 (2004) MR 2085832
19. Willem, M.: Analyse fonctionnelle élémentaire. Cassini, Paris (2003) Zbl pre02243106 MR 1374790