Multiple boundary peak solutions for some singularly perturbed Neumann problems

Changfeng Gui; Juncheng Wei; Matthias Winter

Annales de l'I.H.P. Analyse non linéaire (2000)

  • Volume: 17, Issue: 1, page 47-82
  • ISSN: 0294-1449

How to cite

top

Gui, Changfeng, Wei, Juncheng, and Winter, Matthias. "Multiple boundary peak solutions for some singularly perturbed Neumann problems." Annales de l'I.H.P. Analyse non linéaire 17.1 (2000): 47-82. <http://eudml.org/doc/78487>.

@article{Gui2000,
author = {Gui, Changfeng, Wei, Juncheng, Winter, Matthias},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {multiple boundary spikes; nonlinear elliptic equations},
language = {eng},
number = {1},
pages = {47-82},
publisher = {Gauthier-Villars},
title = {Multiple boundary peak solutions for some singularly perturbed Neumann problems},
url = {http://eudml.org/doc/78487},
volume = {17},
year = {2000},
}

TY - JOUR
AU - Gui, Changfeng
AU - Wei, Juncheng
AU - Winter, Matthias
TI - Multiple boundary peak solutions for some singularly perturbed Neumann problems
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2000
PB - Gauthier-Villars
VL - 17
IS - 1
SP - 47
EP - 82
LA - eng
KW - multiple boundary spikes; nonlinear elliptic equations
UR - http://eudml.org/doc/78487
ER -

References

top
  1. [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. Zbl0847.35047MR1318082
  2. [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. Zbl0793.35033MR1218099
  3. [3] F. Adimurthi Pacella and S.L. Yadava, Characterization of concentration points and L∞-estimates for solutions involving the critical Sobolev exponent, Differential Integral Equations8 (1) (1995) 41-68. Zbl0814.35029MR1296109
  4. [4] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965. Zbl0142.37401MR178246
  5. [5] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math.30 (1978) 33-76. Zbl0407.92014MR511740
  6. [6] E.N. Dancer, A note on asymptotic uniqueness for some nonlinearities which change sign, Rocky Mountain Math. J. , to appear. Zbl0945.35031MR1748710
  7. [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. Zbl0613.35076MR867665
  8. [8] R. Gardner and L.A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh104 A (1986) 53-72. Zbl0625.35030MR877892
  9. [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. Zbl0469.35052
  10. [10] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983. Zbl0562.35001MR737190
  11. [11] C. Gui, Multi-peak solutions for a semilinear Neumann problem, Duke Math. J.84 (1996) 739-769. Zbl0866.35039MR1408543
  12. [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. Zbl0955.35024MR1658569
  13. [13] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations158 (1999) 1-27. Zbl1061.35502MR1721719
  14. [14] B. Helffer and J. Sjöstrand, Multiple wells in the semi-classical limit I, Comm. P.D.E.9 (1984) 337-408. Zbl0546.35053MR740094
  15. [15] J. Jang, On spike solutions of singularly perturbed semilinear Dirichlet problems, J. Differential Equations114 (1994) 370-395. Zbl0812.35008MR1303033
  16. [16] M.K. Kwong, Uniqueness of positive solutions of Δu - u + up = 0 in Rn, Arch. Rational Mech. Anal.105 (1989) 243-266. Zbl0676.35032MR969899
  17. [17] Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E.23 (1998) 487-545. Zbl0898.35004MR1620632
  18. [18] C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations72 (1988) 1-27. Zbl0676.35030MR929196
  19. [19] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol I, Springer, New York, Berlin, Heidelberg, Tokyo, 1972. Zbl0223.35039MR350177
  20. [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. Zbl0785.35041MR1174600
  21. [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. Zbl0754.35042MR1115095
  22. [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. Zbl0796.35056MR1219814
  23. [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. Zbl0843.35006MR1337211
  24. [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. Zbl0838.35009MR1342381
  25. [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. Zbl0702.35228MR970154
  26. [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. Zbl0753.35097MR1065671
  27. [27] X.B. Pan, Condensation of least-energy solutions of a semilinear Neumann problem, J. Partial Differential Equations8 (1995) 1-36. Zbl0814.35039MR1317288
  28. [28] X.B. Pan, Condensation of least-energy solutions: the effect of boundary conditions, Nonlinear Analysis, TMA24 (1995) 195-222. Zbl0826.35037MR1312590
  29. [29] X.B. Pan, Further study on the effect of boundary conditions, J. Differential Equations117 (1995) 446-468. Zbl0832.35050MR1325806
  30. [30] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations39 (1981) 269-290. Zbl0425.34028MR607786
  31. [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. Zbl0784.35035MR1185568
  32. [32] M. Ward, An asymptotic analysis of localized solutions for some reaction-diffusion models in multidimensional domains, Stud. Appl. Math.97 (1996) 103-126. Zbl0932.35059MR1395845
  33. [33] J. Wei, On the construction of single-peaked solutions of a singularly perturbed semilinear Dirichlet problem, J. Differential Equations129 (1996) 315-333. Zbl0865.35011MR1404386
  34. [34] J. Wei, On the effect of the geometry of the domain in a singularly perturbed Dirichlet problem, Differential Integral Equations, to appear. MR1811947
  35. [35] J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Differential Equations134 (1997) 104-133. Zbl0873.35007MR1429093
  36. [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. Zbl0931.35018
  37. [37] J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire15 (1998) 459-492. Zbl0910.35049MR1632937
  38. [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. Zbl0922.35025
  39. [39] K. Yosida, Functional Analysis, 5th ed., Springer, Berlin, 1978. Zbl0365.46001MR500055
  40. [40] E. Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems, Springer, Berlin, 1986. Zbl0583.47050MR816732

Citations in EuDML Documents

top
  1. Manuel del Pino, Michał Kowalczyk, Juncheng Wei, Multi-bump ground states of the Gierer–Meinhardt system in R2
  2. A. Malchiodi, Wei-Ming Ni, Juncheng Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball
  3. Teresa D'Aprile, Angela Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation
  4. Olivier Rey, Juncheng Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Part II : N 4
  5. Tai-Chia Lin, Juncheng Wei, Spikes in two coupled nonlinear Schrödinger equations
  6. Teresa D'Aprile, Juncheng Wei, Clustered solutions around harmonic centers to a coupled elliptic system
  7. Henri Berestycki, Juncheng Wei, On singular perturbation problems with Robin boundary condition
  8. Teresa D’Aprile, Locating the boundary peaks of least-energy solutions to a singularly perturbed Dirichlet problem

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.