References
Alberti, G., Ambrosio, L., Cabré, X.: On a long-standing conjecture of E.De Giorgi: symmetry in 3d for general nonlinearities and a local minimality property. Acta Appl. Math. 65, 9-33 (2002)
Ambrosio, L., Cabré, X.: Entire solutions of semilinear elliptic equations in R 3 and a conjecture of de Giorgi. J. Amer. Math. Soc. 13, 725-739 (2000)
Angenent, S.B., Mallet-Paret, J., Peletier, L.A.: Stable transition layers in a semilinear boundary value problem. J. Diff. Equations 67, 212-242 (1987)
Bates, P., Dancer, E.N., Shi, J.: Multi-spike stationary solutions on the Cahn-Hilliard equation in higher dimension and instability. Adv. Diff. Equations 4, 1-69 (1999)
Berestycki, H., Caffarelli, L., Nirenberg, L.: Further qualitative properties for elliptic equations in unbounded domains. Ann Scuola Norm.Sup.Pisa C1. Sci. 25, 69-94 (1997)
Cao, D., Dancer, E.N., Noussair, E., Yan, S.: On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete and Continuous Dynamical Systems 2, 221-236 (1996)
Cao, D., Küpper, T.: On the existence of the multipeaked solutions to a semilinear Neumann problem. Duke Math. Journal 97, 261-300 (1999)
Clément, P., Peletier, L.A.: On a nonlinear eigenvalue problem occurring in population genetics. Proc. Royal Soc. Edinburgh 100A, 85-101 (1985)
Clément, P., Sweers, G.: Existence and multiplicity results for a semilinear eigenvalue problem. Ann. Scuola Norm. Sup. Pisa 14, 97-121 (1987)
Dancer, E.N.: On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large. Proc. London Math. Soc. 53, 429-452 (1986)
Dancer, E.N., Guo, Z.: Remarks on the stability of sign changing solutions. Tohoko Math. J. 47, 199-225 (1995)
Dancer, E.N., Wei, J.: On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem. Proc. Royal Soc. Edinburgh 127A, 691-701 (1997)
Dancer, E.N., Wei, J.: On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem. J. Diff. Equations 157, 82-101 (1999)
Dancer, E.N., Yan, S.: Multipeak solutions for a singularly perturbed Neumann Problem. Pacific Journal Math. 189, 241-262 (1999)
Dancer, E.N., Yan, S.: Singularly perturbed elliptic problem in exterior domains. J. Diff. Int. Equations 13, 747-777 (2000)
Dancer, E.N., Yan, S.: A singularly perturbed elliptic problem in bounded domains with nontrivial topology. Adv. Diff. Equations 4, 347-368 (1999)
Dancer, E.N., Yan, S.: Interior and boundary peak solutions for a mixed boundary value problem. Indiana Univ. Math. J. 48, 1177-1212 (1999)
Dancer, E.N., Yan, S.: On the existence of multipeak solutions for nonlinear field equations on R N. Discrete and Continuous Dynamical Systems 6, 39-50 (2000)
Dancer, E.N., Yan, S.: On the profile of the changing sign mountain pass solutions for an elliptic problem. Trans. AMS 354, 3573-3600 (2002)
Dancer, E.N., Yan, S.: A minimization problem associated with elliptic systems of FitzHugh-Nagumo type. Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear)
Fife, P.C.: Semilinear elliptic boundary value problems with small parameter. Arch. Rat. Mech. Anal. 52, 205-232 (1973)
Fife, P.C.: Mathematical aspect of reacting and diffusing systems. Lecture Notes in Biomathematics, Vol 28, Springer-Verlag, Berlin/Heidelberg/New York 1979
Fife, P.C., Greenlee, W.W.: Interior transition layers for elliptic boundary value problems with small parameter. Russian Math. Survey 29, 103-131 (1974)
Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 355-369 (1937)
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phy. 68, 209-243 (1979)
De Giorgi, E.: Convergence problems for functionals and operators. In: Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (Rome 1978), 131-188
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin Heidelberg New York Tokyo 1983
Ghoussoub, N., Gui, C.: On a conjecture of De Giorgi and some related problem. Math. Ann. 311, 481-491 (1998)
Ghoussoub, N., Gui, C.: On De Giorgi’s conjecture in dimensions 4 and 5. (preprint)
Gui, C., Wei, J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Diff. Eqns. 158, 1-27 (1999)
Gui, C., Wei, J.: On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems. Canad. J. Math. 152, 522-538 (2000)
Gui, C., Wei, J., Winter, M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 47-82 (2000)
Haldane, J.B.S.: The theory of a cline. J. Genetics 48, 277-284 (1948)
Jang, J.: On the spike solutions of singularly perturbed semilinear Dirichlet problem. J. Diff. Equations 114, 370-395 (1994)
Li, Y.Y.: On a singularly perturbed equation with Neumann boundary condition. Comm. P.D.E. 23, 487-545 (1998)
Modica, L.: Gradient theory of phase transitions and minimal interface criteria. Arch. Rational Mech. Anal. 98, 123-142 (1987)
Ni, W.M., Takagi, I.: On the shape of the least energy solution to a semilinear Neumann problem. Comm. Pure Appl. Math. 41, 819-851 (1991)
Ni, W.M., Takagi, I.: Locating the peaks of least energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247-281 (1993)
Ni, W.M., Takagi, I., Wei, J.: On the locations and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem, intermediate solution. Duke Math. J. 94, 597-618 (1998)
Ni, W.M., Wei, J.: On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. Comm. Pure Appl. Math. 48, 731-768 (1995)
Noussair, E.S., Yan, S.: On the multipeak solutions of a nonlinear elliptic problem. J. London Math. Soc. 61, 213-227 (2000)
Peletier, L.A., Serrin, J.: Uniqueness of positive solutions of semilinear equations in R n. Arch. Rat. Mech. Anal. 81, 181-197 (1983)
Rey, O.: The role of the Green’s function in a non-linear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89, 1-52 (1990)
Sweers, G.: A sign-changing global minimizer on a convex domain. In: Bandlek, C. et al. (ed.) Progress in partial differential equations, pp 251-258. Pitman 1992
Sternberg, P.: The effect of a singular perturbation on nonconvex variational problem. Arch. rational Mech. Anal. 101, 202-260 (1988)
Wei, J., Winter, M.: Stationary solutions for the Cahn-Hilliard equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 459-492 (1998)
Yan, S.: On the number of interior multipeak solutions for singularly perturbed Neumann problems. Topological Methods in Nonlinear Anal. 12, 61-78 (1999)
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Received: 29 May 2002, Accepted: 12 June 2003, Published online: 4 September 2003
This work was supported by the ARC
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Dancer, E., Yan, S. Construction of various types of solutions for an elliptic problem. Cal Var 20, 93–118 (2004). https://doi.org/10.1007/s00526-003-0229-6
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DOI: https://doi.org/10.1007/s00526-003-0229-6