Monotonicity constraints and supercritical Neumann problems
Serra, Enrico ; Tilli, Paolo
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 63-74 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

On démontre l'existence d'une solution positive et radialement croissante pour un problème de Neumann semilinéaire sur une boule. Aucune restriction de croissance n'est imposée sur la nonlinéarité. La méthode indroduit des contraintes de monotonie qui simplifient la preuve de l'existence d'un minimum pour la fonctionnelle associée à l'équation. Une attention particulière est consacrée à la preuve de la validité de l'équation d'Euler.

We prove the existence of a positive and radially increasing solution for a semilinear Neumann problem on a ball. No growth conditions are imposed on the nonlinearity. The method introduces monotonicity constraints which simplify the existence of a minimizer for the associated functional. Special care must be employed to establish the validity of the Euler equation.

Publié le : 2011-01-01
DOI : https://doi.org/10.1016/j.anihpc.2010.10.003
Classification:  35J60,  58E30 
@article{AIHPC_2011__28_1_63_0,
     author = {Serra, Enrico and Tilli, Paolo},
     title = {Monotonicity constraints and supercritical Neumann problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {28},
     year = {2011},
     pages = {63-74},
     doi = {10.1016/j.anihpc.2010.10.003},
     mrnumber = {2765510},
     zbl = {1209.35044},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2011__28_1_63_0}
}

Serra, Enrico; Tilli, Paolo. Monotonicity constraints and supercritical Neumann problems. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) pp. 63-74. doi : 10.1016/j.anihpc.2010.10.003. http://gdmltest.u-ga.fr/item/AIHPC_2011__28_1_63_0/

[1] Adimurthi, S.L. Yadava, Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents, Arch. Ration. Mech. Anal. 115 no. 3 (1991), 275-296 | MR 1106295 | Zbl 0839.35041

[2] Adimurthi, S.L. Yadava, On a conjecture of Lin–Ni for a semilinear Neumann problem, Trans. Amer. Math. Soc. 336 no. 2 (1993), 631-637 | MR 1156299 | Zbl 0787.35030

[3] V. Barutello, S. Secchi, E. Serra, A note on the radial solutions for the supercritical Hònon equation, J. Math. Anal. Appl. 341 no. 1 (2008), 720-728 | MR 2394119 | Zbl 1135.35031

[4] A. Malchiodi, W.-M. Ni, J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 2 (2005), 143-163 | Numdam | MR 2124160 | Zbl 1207.35141

[5] C.-S. Lin, Locating the peaks of solutions via the maximum principle. I. The Neumann problem, Comm. Pure Appl. Math. 54 no. 9 (2001), 1065-1095 | MR 1835382 | Zbl 1035.35039

[6] C.-S. Lin, W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of Variations and Partial Differential Equations, Trento, 1986, Lecture Notes in Math. vol. 1340, Springer, Berlin (1988), 160-174

[7] C.-S. Lin, W.-M. Ni, I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 no. 1 (1988), 1-27 | MR 929196 | Zbl 0676.35030

[8] R. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 no. 1 (1979), 19-30 | MR 547524 | Zbl 0417.58007

[9] M. Struwe, On a critical point theory for minimal surfaces spanning a wire in R n , J. Reine Angew. Math. 349 (1984), 1-23 | MR 743962 | Zbl 0521.49028

[10] M. Zhu, Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differential Equations 154 no. 2 (1999), 284-317 | MR 1691074 | Zbl 0927.35041