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.
Mots-clés : Constrained variational problems, Supercritical elliptic equations, Neumann problem
@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},
pages = {63--74},
publisher = {Elsevier},
volume = {28},
number = {1},
year = {2011},
doi = {10.1016/j.anihpc.2010.10.003},
mrnumber = {2765510},
zbl = {1209.35044},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.10.003/}
}
TY - JOUR AU - Serra, Enrico AU - Tilli, Paolo TI - Monotonicity constraints and supercritical Neumann problems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2011 SP - 63 EP - 74 VL - 28 IS - 1 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2010.10.003/ DO - 10.1016/j.anihpc.2010.10.003 LA - en ID - AIHPC_2011__28_1_63_0 ER -
%0 Journal Article %A Serra, Enrico %A Tilli, Paolo %T Monotonicity constraints and supercritical Neumann problems %J Annales de l'I.H.P. Analyse non linéaire %D 2011 %P 63-74 %V 28 %N 1 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2010.10.003/ %R 10.1016/j.anihpc.2010.10.003 %G en %F 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) no. 1, pp. 63-74. doi : 10.1016/j.anihpc.2010.10.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.10.003/
[1] , , 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 | Zbl
[2] , , On a conjecture of Lin–Ni for a semilinear Neumann problem, Trans. Amer. Math. Soc. 336 no. 2 (1993), 631-637 | MR | Zbl
[3] , , , A note on the radial solutions for the supercritical Hònon equation, J. Math. Anal. Appl. 341 no. 1 (2008), 720-728 | MR | Zbl
[4] , , , Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 no. 2 (2005), 143-163 | EuDML | Numdam | MR | Zbl
[5] , Locating the peaks of solutions via the maximum principle. I. The Neumann problem, Comm. Pure Appl. Math. 54 no. 9 (2001), 1065-1095 | MR | Zbl
[6] , , 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] , , , Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 no. 1 (1988), 1-27 | MR | Zbl
[8] , The principle of symmetric criticality, Comm. Math. Phys. 69 no. 1 (1979), 19-30 | MR | Zbl
[9] , On a critical point theory for minimal surfaces spanning a wire in , J. Reine Angew. Math. 349 (1984), 1-23 | EuDML | MR | Zbl
[10] , Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differential Equations 154 no. 2 (1999), 284-317 | MR | Zbl
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