Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : N4
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 4, pp. 459-484.
@article{AIHPC_2005__22_4_459_0,
     author = {Rey, Olivier and Wei, Juncheng},
     title = {Blowing up solutions for an elliptic {Neumann} problem with sub- or supercritical nonlinearity. {Part} {II} : $N\ge 4$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {459--484},
     publisher = {Elsevier},
     volume = {22},
     number = {4},
     year = {2005},
     doi = {10.1016/j.anihpc.2004.07.004},
     mrnumber = {2145724},
     zbl = {02191850},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.004/}
}
TY  - JOUR
AU  - Rey, Olivier
AU  - Wei, Juncheng
TI  - Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : $N\ge 4$
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2005
SP  - 459
EP  - 484
VL  - 22
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.004/
DO  - 10.1016/j.anihpc.2004.07.004
LA  - en
ID  - AIHPC_2005__22_4_459_0
ER  - 
%0 Journal Article
%A Rey, Olivier
%A Wei, Juncheng
%T Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : $N\ge 4$
%J Annales de l'I.H.P. Analyse non linéaire
%D 2005
%P 459-484
%V 22
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.004/
%R 10.1016/j.anihpc.2004.07.004
%G en
%F AIHPC_2005__22_4_459_0
Rey, Olivier; Wei, Juncheng. Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : $N\ge 4$. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 4, pp. 459-484. doi : 10.1016/j.anihpc.2004.07.004. http://www.numdam.org/articles/10.1016/j.anihpc.2004.07.004/

[1] Adimurthi , Mancini G., The Neumann problem for elliptic equations with critical nonlinearity, “A tribute in honour of G. Prodi”, Scuola Norm. Sup. Pisa (1991) 9-25. | Zbl

[2] Adimurthi , Mancini G., Geometry and topology of the boundary in the critical Neumann problem, J. Reine Angew. Math. 456 (1994) 1-18. | EuDML | MR | Zbl

[3] Adimurthi , Pacella F., Yadava S.L., Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal. 113 (1993) 318-350. | MR | Zbl

[4] Bahri A., Critical Points at Infinity in Some Variational Problems, Pitman Res. Notes Math. Ser., vol. 182, Longman, 1989. | MR | Zbl

[5] Bates P., Fusco G., Equilibria with many nuclei for the Cahn-Hilliard equation, J. Differential Equations 160 (2000) 283-356. | MR | Zbl

[6] Caffarelli L., Gidas B., Spruck J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989) 271-297. | MR | Zbl

[7] Cerami G., Wei J., Multiplicity of multiple interior peaks solutions for some singularly perturbed Neumann problems, Intern. Math. Res. Notes 12 (1998) 601-626. | MR | Zbl

[8] Dancer E.N., Yan S., Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999) 241-262. | MR | Zbl

[9] Del Pino M., Felmer P., Musso M., Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations 16 (2003) 113-145. | MR | Zbl

[10] Grossi M., Pistoia A., On the effect of critical points of distance function in superlinear elliptic problems, Adv. Differential Equations 5 (2000) 1397-1420. | MR | Zbl

[11] Grossi M., Pistoia A., Wei J., Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2000) 143-175. | MR | Zbl

[12] Gierer A., Meinhardt H., A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972) 30-39.

[13] Gui C., Multi-peak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996) 739-769. | MR | Zbl

[14] Gui C., Lin C.S., Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math. 546 (2002) 201-235. | MR | Zbl

[15] Gui C., Wei J., Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999) 1-27. | MR | Zbl

[16] C. Gui, J. Wei, On the existence of arbitrary number of bubbles for some semilinear elliptic equations with critical Sobolev exponent, in press.

[17] Gui C., Wei J., On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000) 522-538. | MR | Zbl

[18] Gui C., Wei J., Winter M., Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 47-82. | Numdam | MR | Zbl

[19] Khenissy S., Rey O., A criterion for existence of solutions to the supercritical Bahri-Coron's problem, Houston J. Math. 30 (2004) 587-613. | MR | Zbl

[20] Kowalczyk M., Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and quasi-invariant manifold, Duke Math. J. 98 (1999) 59-111. | MR | Zbl

[21] Li Y., Ni W.-M., On conformal scalar curvature equation in R n , Duke Math. J. 57 (1988) 895-924. | MR | Zbl

[22] Li Y.Y., On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998) 487-545. | MR | Zbl

[23] Lin C.S., Ni W.M., On the Diffusion Coefficient of a Semilinear Neumann Problem, Lecture Notes in Math., vol. 1340, Springer, New York, 1986. | MR | Zbl

[24] Lin C.S., Ni W.N., Takagi I., Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988) 1-27. | MR | Zbl

[25] Mcowen R., The behavior of the Laplacian on weighted Sobolev spaces, Comm. Pure Appl. Math. 32 (1979) 783-795. | MR | Zbl

[26] Maier-Paape S., Schmitt K., Wang Z.Q., On Neumann problems for semilinear elliptic equations with critical nonlinearity: existence and symmetry of multi-peaked solutions, Comm. Partial Differential Equations 22 (1997) 1493-1527. | MR | Zbl

[27] Ni W.-M., Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998) 9-18. | MR | Zbl

[28] Ni W.N., Pan X.B., Takagi I., Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992) 1-20. | MR | Zbl

[29] Ni W.N., Takagi I., On the shape of least-energy solutions to a semi-linear problem Neumann problem, Comm. Pure Appl. Math. 44 (1991) 819-851. | MR | Zbl

[30] Ni W.M., Takagi I., Locating the peaks of least-energy solutions to a semi-linear Neumann problem, Duke Math. J. 70 (1993) 247-281. | MR | Zbl

[31] Rey O., The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990) 1-52. | MR | Zbl

[32] Rey O., An elliptic Neumann problem with critical nonlinearity in three dimensional domains, Comm. Contemp. Math. 1 (1999) 405-449. | MR | Zbl

[33] Rey O., The question of interior blow-up points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl. 81 (2002) 655-696. | MR | Zbl

[34] O. Rey, J. Wei, Blow-up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity, I: N=3, J. Funct. Anal., in press. | Zbl

[35] Wang X.J., Neumann problem of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991) 283-310. | MR | Zbl

[36] Wang Z.Q., The effect of domain geometry on the number of positive solutions of Neumann problems with critical exponents, Differential Integral Equations 8 (1995) 1533-1554. | MR | Zbl

[37] Wang Z.Q., High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 1003-1029. | MR | Zbl

[38] Wang Z.Q., Construction of multi-peaked solution for a nonlinear Neumann problem with critical exponent, J. Nonlinear Anal. 27 (1996) 1281-1306. | MR | Zbl

[39] Wang X., Wei J., On the equation Δu+Kx,u n+2 n-2±ϵ 2 =0 in R n , Rend. Circ. Mat. Palermo 2 (1995) 365-400. | Zbl

[40] Wei J., On the interior spike layer solutions of singularly perturbed semilinear Neumann problems, Tohoku Math. J. 50 (1998) 159-178. | MR | Zbl

[41] J. Wei, X. Xu, Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in R 3 , Pacific J. Math., in press. | MR | Zbl

[42] Wei J., Winter M., Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré, Anal. Non Linéaire 15 (1998) 459-482. | Numdam | MR | Zbl

[43] Yan S., On the number of interior multipeak solutions for singularly perturbed Neumann problems, Topol. Methods Nonlinear Anal. 12 (1998) 61-78. | MR | Zbl

[44] Zhu M., Uniqueness results through a priori estimates, I. A three dimensional Neumann problem, J. Differential Equations 154 (1999) 284-317. | MR | Zbl

Cité par Sources :