Multiple boundary peak solutions for some singularly perturbed Neumann problems
Annales de l'I.H.P. Analyse non linéaire, Volume 17 (2000) no. 1, p. 47-82
@article{AIHPC_2000__17_1_47_0,
author = {Gui, Changfeng and Wei, Juncheng and Winter, Matthias},
title = {Multiple boundary peak solutions for some singularly perturbed Neumann problems},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Gauthier-Villars},
volume = {17},
number = {1},
year = {2000},
pages = {47-82},
zbl = {0944.35020},
mrnumber = {1743431},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2000__17_1_47_0}
}

Gui, Changfeng; Wei, Juncheng; Winter, Matthias. Multiple boundary peak solutions for some singularly perturbed Neumann problems. Annales de l'I.H.P. Analyse non linéaire, Volume 17 (2000) no. 1, pp. 47-82. http://www.numdam.org/item/AIHPC_2000__17_1_47_0/

[1] G. Adimurthi Mancinni and S.L. Yadava, The role of mean curvature in a semilinear Neumann problem involving the critical Sobolev exponent, Comm. P.D.E. 20 (1995) 591-631. | MR 1318082 | Zbl 0847.35047

[2] F. Adimurthi Pacella and S.L. Yadava, 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 1218099 | Zbl 0793.35033

[3] F. Adimurthi Pacella and S.L. Yadava, Characterization of concentration points and L∞-estimates for solutions involving the critical Sobolev exponent, Differential Integral Equations 8 (1) (1995) 41-68. | MR 1296109 | Zbl 0814.35029

[4] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965. | MR 178246 | Zbl 0142.37401

[5] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978) 33-76. | MR 511740 | Zbl 0407.92014

[6] E.N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign, Rocky Mountain Math. J. , to appear. | MR 1748710

[7] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397-408. | MR 867665 | Zbl 0613.35076

[8] R. Gardner and L.A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh 104 A (1986) 53-72. | MR 877892 | Zbl 0625.35030

[9] B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, in: Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies, Vol. 7A, Academic Press, New York, 1981, pp. 369-402. | Zbl 0469.35052

[10] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. | MR 737190 | Zbl 0562.35001

[11] C. Gui, Multi-peak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996) 739-769. | MR 1408543 | Zbl 0866.35039

[12] C. Gui and N. Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998) 443-474. | MR 1658569 | Zbl 0955.35024

[13] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999) 1-27. | MR 1721719 | Zbl 1061.35502

[14] B. Helffer and J. Sjöstrand, Multiple wells in the semi-classical limit I, Comm. P.D.E. 9 (1984) 337-408. | MR 740094 | Zbl 0546.35053

[15] J. Jang, On spike solutions of singularly perturbed semilinear Dirichlet problems, J. Differential Equations 114 (1994) 370-395. | MR 1303033 | Zbl 0812.35008

[16] M.K. Kwong, Uniqueness of positive solutions of Δu - u + up = 0 in Rn, Arch. Rational Mech. Anal. 105 (1989) 243-266. | MR 969899 | Zbl 0676.35032

[17] Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E. 23 (1998) 487-545. | MR 1620632 | Zbl 0898.35004

[18] C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations 72 (1988) 1-27. | MR 929196 | Zbl 0676.35030

[19] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol I, Springer, New York, Berlin, Heidelberg, Tokyo, 1972. | MR 350177 | Zbl 0223.35039

[20] W.-M. Ni, X. Pan and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992) 1-20. | MR 1174600 | Zbl 0785.35041

[21] W.-M. Ni and I. Takagi, On the shape of least energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 41 (1991) 819-851. | MR 1115095 | Zbl 0754.35042

[22] W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993) 247-281. | MR 1219814 | Zbl 0796.35056

[23] W.-M. Ni and I. Takagi, Point-condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Industrial Appl. Math. 12 (1995) 327-365. | MR 1337211 | Zbl 0843.35006

[24] W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995) 731-768. | MR 1342381 | Zbl 0838.35009

[25] Y.G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. P.D.E. 13 (12) (1988) 1499- 1519. | MR 970154 | Zbl 0702.35228

[26] Y.G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple-well potentials, Comm. Math. Phys. 131 (1990) 223-253. | MR 1065671 | Zbl 0753.35097

[27] X.B. Pan, Condensation of least-energy solutions of a semilinear Neumann problem, J. Partial Differential Equations 8 (1995) 1-36. | MR 1317288 | Zbl 0814.35039

[28] X.B. Pan, Condensation of least-energy solutions: the effect of boundary conditions, Nonlinear Analysis, TMA 24 (1995) 195-222. | MR 1312590 | Zbl 0826.35037

[29] X.B. Pan, Further study on the effect of boundary conditions, J. Differential Equations 117 (1995) 446-468. | MR 1325806 | Zbl 0832.35050

[30] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981) 269-290. | MR 607786 | Zbl 0425.34028

[31] Z.-Q. Wang, On the existence of multiple single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal. 120 (1992) 375-399. | MR 1185568 | Zbl 0784.35035

[32] M. Ward, An asymptotic analysis of localized solutions for some reaction-diffusion models in multidimensional domains, Stud. Appl. Math. 97 (1996) 103-126. | MR 1395845 | Zbl 0932.35059

[33] J. Wei, On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996) 315-333. | MR 1404386 | Zbl 0865.35011

[34] J. Wei, On the effect of the geometry of the domain in a singularly perturbed Dirichlet problem, Differential Integral Equations, to appear. | MR 1811947

[35] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Differential Equations 134 (1997) 104-133. | MR 1429093 | Zbl 0873.35007

[36] J. Wei, On the construction of single interior peak solutions for a singularly perturbed Neumann problem, in: Partial Differential Equations: Theory and Numerical solution; CRC Press LLC, 1998, pp. 336-349. | Zbl 0931.35018

[37] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 459-492. | Numdam | MR 1632937 | Zbl 0910.35049

[38] J. Wei and M. Winter, Multiple boundary spike solutions for a wide class of singular perturbation problems, J. London Math. Soc. 59 (2) (1999) 585-606. | Zbl 0922.35025

[39] K. Yosida, Functional Analysis, 5th ed., Springer, Berlin, 1978. | MR 500055 | Zbl 0365.46001

[40] E. Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems, Springer, Berlin, 1986. | MR 816732 | Zbl 0583.47050