Super-critical boundary bubbling in a semilinear Neumann problem

del Pino, Manuel; Musso, Monica; Pistoia, Angela

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 1, pp. 45-82.

[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. | MR | 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 , Mancini G., Yadava S.L., The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations 20 (3-4) (1995) 591-631. | MR | Zbl

[4] 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

[5] Adimurthi , Pacella F., Yadava S.L., Characterization of concentration points and $L^{\infty }$-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Differential Integral Equations 8 (1) (1995) 41-68. | MR | Zbl

[6] Cao D., Noussair E.S., The effect of geometry of the domain boundary in an elliptic Neumann problem, Adv. Differential Equations 6 (8) (2001) 931-958. | MR | Zbl

[7] Dancer E.N., Yan S., Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (2) (1999) 241-262. | MR | Zbl

[8] Del Pino M., Dolbeault J., Musso M., “Bubble-tower” radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differential Equations 193 (2) (2003) 280-306. | Zbl

[9] Del Pino M., Felmer P., Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (3) (1999) 883-898. | MR | Zbl

[10] Del Pino M., Felmer P., Musso M., Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. PDE 16 (2) (2003) 113-145. | MR | Zbl

[11] Del Pino M., Felmer P., Wei J., On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1) (1999) 63-79. | MR | Zbl

[12] Fowler R.H., Further studies on Emden's and similar differential equations, Quart. J. Math. 2 (1931) 259-288. | Zbl

[13] Grossi M., A class of solutions for the Neumann problem $-\Delta u+\lambda u=u^{(N+2)/(N-2)}$, Duke Math. J. 79 (2) (1995) 309-334. | MR | Zbl

[14] Grossi M., Pistoia A., Wei J., Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2) (2000) 143-175. | MR | Zbl

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

[16] Gui C., Ghoussoub N., Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (3) (1998) 443-474. | MR | Zbl

[17] 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

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

[19] Kowalczyk M., Multiple spike layers in the shadow Gierer-Meinhardt system: existence of equilibria and the quasi-invariant manifold, Duke Math. J. 98 (1) (1999) 59-111. | MR | Zbl

[20] Li Y.Y., On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (3-4) (1998) 487-545. | MR | Zbl

[21] Li Y.Y., Prescribing scalar curvature on $S^n$ and related problems, part I, J. Differential Equations 120 (1996) 541-597. | MR | Zbl

[22] Y.Y. Li, L. Zhang, Liouville and Harnack type theorems for semilinear elliptic equations, preprint.

[23] Lin C.-S., Locating the peaks of solutions via the maximum principle, I. The Neumann problem, Comm. Pure Appl. Math. 54 (2001) 1065-1095. | MR | Zbl

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

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

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

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

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

[29] Rey O., Boundary effect for an elliptic Neumann problem with critical nonlinearity, Comm. in PDE 22 (1997) 1055-1139. | MR | Zbl

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

[31] O. Rey, J. Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity, part I: $N=3$, J. Funct. Anal., submitted for publication. | Zbl

[32] Wang X.J., Neumann problem of semilinear elliptic equations involving critical Sobolev exponent, J. Differential Equations 93 (1991) 283-301. | MR | Zbl

[33] 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

[34] Wei J., On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1) (1997) 104-133. | MR | Zbl