Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Dancer, E.

doi:10.1090/S0002-9947-04-03543-3

Stable and finite Morse index solutions on $\mathbf {R}^n$ or on bounded domains with small diffusion
HTML articles powered by AMS MathViewer

by E. N. Dancer PDF

Trans. Amer. Math. Soc. 357 (2005), 1225-1243 Request permission

Abstract:

In this paper, we study bounded solutions of $- \Delta u = f (u)$ on $\mathbf {R}^n$ (where $n = 2$ and sometimes $n = 3$) and show that, for most $f$’s, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of $- \epsilon ^2 \Delta u = f (u)$ on $\Omega$ with Dirichlet or Neumann boundary conditions for small $\epsilon$.

References

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. MR 1843784, DOI 10.1023/A:1010602715526
Luigi Ambrosio and Xavier Cabré, Entire solutions of semilinear elliptic equations in $\mathbf R^3$ and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000), no. 4, 725–739. MR 1775735, DOI 10.1090/S0894-0347-00-00345-3
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. MR 1177482, DOI 10.1002/cpa.3160450908
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. MR 1655510
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. MR 1395408, DOI 10.1215/S0012-7094-96-08117-X
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. MR 480282, DOI 10.1016/0022-0396(78)90033-5
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. MR 937538
Edward Norman Dancer, New solutions of equations on $\mathbf R^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 3-4, 535–563 (2002). MR 1896077
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. MR 868453, DOI 10.1112/plms/s3-53.3.429
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. MR 949628, DOI 10.1016/0022-0396(88)90021-6
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. MR 1350350, DOI 10.12775/TMNA.1995.009
E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), no. 3, 425–434. MR 1190345, DOI 10.1017/S0004972700012089
E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math. 371 (1986), 46–66. MR 859319, DOI 10.1515/crll.1986.371.46
E. N. Dancer, Infinitely many turning points for some supercritical problems, Ann. Mat. Pura Appl. (4) 178 (2000), 225–233. MR 1849387, DOI 10.1007/BF02505896
E. N. Dancer, Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z. 229 (1998), no. 3, 475–491. MR 1658565, DOI 10.1007/PL00004666
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. MR 1329521, DOI 10.2748/tmj/1178225592
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. MR 1116922
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. MR 1465415, DOI 10.1017/S0308210500023775
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. MR 1710015, DOI 10.1006/jdeq.1998.3619
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. MR 1911512, DOI 10.1090/S0002-9947-02-03026-X
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. MR 1758878, DOI 10.12775/TMNA.1999.020
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. MR 1757072, DOI 10.1512/iumj.1999.48.1827
E. N. Dancer and Klaus Schmitt, On positive solutions of semilinear elliptic equations, Proc. Amer. Math. Soc. 101 (1987), no. 3, 445–452. MR 908646, DOI 10.1090/S0002-9939-1987-0908646-2
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. MR 1736974, DOI 10.1512/iumj.1999.48.1596
N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI 10.1007/s002080050196
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
Changfeng Gui and Juncheng Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations 158 (1999), no. 1, 1–27. MR 1721719, DOI 10.1016/S0022-0396(99)80016-3
Peter Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations 6 (1981), no. 8, 951–961. MR 619753, DOI 10.1080/03605308108820200
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. MR 727256, DOI 10.1090/S0002-9939-1984-0727256-0
Jaeduck Jang, On spike solutions of singularly perturbed semilinear Dirichlet problem, J. Differential Equations 114 (1994), no. 2, 370–395. MR 1303033, DOI 10.1006/jdeq.1994.1154
Robert V. Kohn and Peter Sternberg, Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 69–84. MR 985990, DOI 10.1017/S0308210500025026
Yi Li and Wei-Ming Ni, Radial symmetry of positive solutions of nonlinear elliptic equations in $\textbf {R}^n$, Comm. Partial Differential Equations 18 (1993), no. 5-6, 1043–1054. MR 1218528, DOI 10.1080/03605309308820960
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. MR 1639546, DOI 10.1215/S0012-7094-98-09424-8
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. MR 1342381, DOI 10.1002/cpa.3160480704
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. MR 943809, DOI 10.1017/S0308210500014724
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. MR 1404386, DOI 10.1006/jdeq.1996.0120
Juncheng Wei, On the interior spike solutions for some singular perturbation problems, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 4, 849–874. MR 1635448, DOI 10.1017/S030821050002182X
Juncheng Wei, On the effect of domain geometry in singular perturbation problems, Differential Integral Equations 13 (2000), no. 1-3, 15–45. MR 1811947
Shusen Yan, On the number of interior multipeak solutions for singularly perturbed Neumann problems, Topol. Methods Nonlinear Anal. 12 (1998), no. 1, 61–78. MR 1677747, DOI 10.12775/TMNA.1998.028
E.N. Dancer, Yihong Du, “Some remarks on Liouville type results for quasilinear elliptic equations”, Proc Amer Math Soc 131 (2003), 1891-1899.
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. MR 1198672, DOI 10.1007/BF00916424
Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), no. 2, 401–454. MR 555661, DOI 10.2977/prims/1195188180
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. MR 1363002, DOI 10.1017/S0308210500030493
Junping Shi, Saddle solutions of the balanced bistable diffusion equation, Comm. Pure Appl. Math. 55 (2002), no. 7, 815–830. MR 1894156, DOI 10.1002/cpa.3027.abs