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Stable and finite Morse index solutions on \(\mathbb R^n\) or on bounded domains with small diffusion. (English) Zbl 1145.35369
The goal of this paper is to study the equation \[ -\Delta u= f(u) \] on \(\mathbb R^d\) (or a half space). The author is interested in solutions which are weakly stable or have finite Morse index. He shows that for \(d= 2\) and sometimes for \(d=3\) these solutions are very simple and easy to understand. As an application of these ideas, the author presents a number of results on the solutions of \[ -\varepsilon^2\Delta u= f(u)\quad\text{in }\Omega, \] where \(\Omega\) is a bounded open set in \(\mathbb R^2\) (sometimes in \(\mathbb R^3\)) with Neumann or Dirichlet boundary conditions.

MSC:
35J60 Nonlinear elliptic equations
35B25 Singular perturbations in context of PDEs
35B35 Stability in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré, On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math. 65 (2001), no. 1-3, 9 – 33. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. · Zbl 1121.35312
[2] Luigi Ambrosio and Xavier Cabré, Entire solutions of semilinear elliptic equations in \?³ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725 – 739. · Zbl 0968.35041
[3] A. Bahri and P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), no. 9, 1205 – 1215. · Zbl 0801.35026
[4] Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69 – 94 (1998). Dedicated to Ennio De Giorgi. · Zbl 1079.35513
[5] H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains. I, Duke Math. J. 81 (1996), no. 2, 467 – 494. A celebration of John F. Nash, Jr. · Zbl 0860.35004
[6] Richard G. Casten and Charles J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), no. 2, 266 – 273. · Zbl 0338.35055
[7] Philippe Clément and Guido Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 97 – 121. · Zbl 0662.35045
[8] Edward Norman Dancer, New solutions of equations on \?\(^{n}\), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 3-4, 535 – 563 (2002). · Zbl 1025.35009
[9] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc. (3) 53 (1986), no. 3, 429 – 452. · Zbl 0572.35040
[10] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Differential Equations 74 (1988), no. 1, 120 – 156. · Zbl 0662.34025
[11] E. N. Dancer, On positive solutions of some singularly perturbed problems where the nonlinearity changes sign, Topol. Methods Nonlinear Anal. 5 (1995), no. 1, 141 – 175. Contributions dedicated to Ky Fan on the occasion of his 80th birthday. · Zbl 0835.35013
[12] E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), no. 3, 425 – 434. · Zbl 0777.35005
[13] E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math. 371 (1986), 46 – 66. · Zbl 0597.47034
[14] E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000), 225 – 233. · Zbl 1030.35073
[15] E. N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z. 229 (1998), no. 3, 475 – 491. · Zbl 0933.35068
[16] E. Norman Dancer and Zong Ming Guo, Some remarks on the stability of sign changing solutions, Tohoku Math. J. (2) 47 (1995), no. 2, 199 – 225. · Zbl 0837.35020
[17] E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math. 419 (1991), 125 – 139. · Zbl 0728.58018
[18] E. N. Dancer and Juncheng Wei, On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 691 – 701. · Zbl 0882.35052
[19] E. N. Dancer and Juncheng Wei, On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 157 (1999), no. 1, 82 – 101. · Zbl 1087.35507
[20] E. N. Dancer and Shusen Yan, On the profile of the changing sign mountain pass solutions for an elliptic problem, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3573 – 3600. · Zbl 1109.35041
[21] E. Norman Dancer and Shusen Yan, Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem, Topol. Methods Nonlinear Anal. 14 (1999), no. 1, 1 – 38. · Zbl 0958.35054
[22] E. N. Dancer and Shusen Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), no. 4, 1177 – 1212. · Zbl 0948.35055
[23] E. N. Dancer and Klaus Schmitt, On positive solutions of semilinear elliptic equations, Proc. Amer. Math. Soc. 101 (1987), no. 3, 445 – 452. · Zbl 0661.35031
[24] Manuel Del Pino and Patricio L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J. 48 (1999), no. 3, 883 – 898. · Zbl 0932.35080
[25] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481 – 491. · Zbl 0918.35046
[26] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[27] Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1 – 27. · Zbl 1061.35502
[28] Peter Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations 6 (1981), no. 8, 951 – 961. · Zbl 0468.35073
[29] Helmut Hofer, A note on the topological degree at a critical point of mountainpass-type, Proc. Amer. Math. Soc. 90 (1984), no. 2, 309 – 315. · Zbl 0545.58015
[30] Jaeduck Jang, On spike solutions of singularly perturbed semilinear Dirichlet problem, J. Differential Equations 114 (1994), no. 2, 370 – 395. · Zbl 0812.35008
[31] Robert V. Kohn and Peter Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 69 – 84. · Zbl 0676.49011
[32] Yi Li and Wei-Ming Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in \?\(^{n}\), Comm. Partial Differential Equations 18 (1993), no. 5-6, 1043 – 1054. · Zbl 0788.35042
[33] Wei-Ming Ni, Izumi Takagi, and Juncheng Wei, On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: intermediate solutions, Duke Math. J. 94 (1998), no. 3, 597 – 618. · Zbl 0946.35007
[34] Wei-Ming Ni and Juncheng Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), no. 7, 731 – 768. · Zbl 0838.35009
[35] Guido Sweers, On the maximum of solutions for a semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 3-4, 357 – 370. · Zbl 0681.35013
[36] Juncheng Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations 129 (1996), no. 2, 315 – 333. · Zbl 0865.35011
[37] Juncheng Wei, On the interior spike solutions for some singular perturbation problems, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 4, 849 – 874. · Zbl 0944.35021
[38] Juncheng Wei, On the effect of domain geometry in singular perturbation problems, Differential Integral Equations 13 (2000), no. 1-3, 15 – 45. · Zbl 0970.35034
[39] Shusen Yan, On the number of interior multipeak solutions for singularly perturbed Neumann problems, Topol. Methods Nonlinear Anal. 12 (1998), no. 1, 61 – 78. · Zbl 0929.35056
[40] E.N. Dancer, Yihong Du, “Some remarks on Liouville type results for quasilinear elliptic equations”, Proc Amer Math Soc 131 (2003), 1891-1899. · Zbl 1076.35038
[41] Ha Dang, Paul C. Fife, and L. A. Peletier, Saddle solutions of the bistable diffusion equation, Z. Angew. Math. Phys. 43 (1992), no. 6, 984 – 998. · Zbl 0764.35048
[42] Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), no. 2, 401 – 454. · Zbl 0445.35063
[43] M. Schatzman, On the stability of the saddle solution of Allen-Cahn’s equation, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1241 – 1275. · Zbl 0852.35020
[44] Junping Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math. 55 (2002), no. 7, 815 – 830. · Zbl 1124.35316
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