Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity

Byeon, Jaeyoung

doi:10.1090/S0002-9947-09-04746-1

Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity
HTML articles powered by AMS MathViewer

by Jaeyoung Byeon PDF

Trans. Amer. Math. Soc. 362 (2010), 1981-2001 Request permission

Abstract:

Let $\Omega$ be a bounded domain in $\mathbf {R}^n,$ $n \ge 3,$ with a boundary $\partial \Omega \in C^2.$ We consider the following singularly perturbed nonlinear elliptic problem on $\Omega$: \[ \varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega , \] where the nonlinearity $f$ is of subcritical growth. Under rather strong conditions on $f,$ it has been known that for small $\varepsilon > 0,$ there exists a mountain pass solution $u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function $d$ from $\partial \Omega$ as $\varepsilon \to 0.$ In this paper, we construct a solution $u_\varepsilon$ of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on $f$, which we believe to be almost optimal.

References

Frederick J. Almgren Jr. and Elliott H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), no. 4, 683–773. MR 1002633, DOI 10.1090/S0894-0347-1989-1002633-4
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
Jaeyoung Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1731–1769. MR 1469588, DOI 10.1080/03605309708821317
Jaeyoung Byeon, Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems, J. Differential Equations 217 (2005), no. 2, 257–281. MR 2168823, DOI 10.1016/j.jde.2005.07.008
Jaeyoung Byeon and Louis Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185–200. MR 2317788, DOI 10.1007/s00205-006-0019-3
Jaeyoung Byeon and Louis Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dyn. Syst. 19 (2007), no. 2, 255–269. MR 2335747, DOI 10.3934/dcds.2007.19.255
Jaeyoung Byeon and Louis Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), no. 2, 185–200. MR 2317788, DOI 10.1007/s00205-006-0019-3
Henri Berestycki, Thierry Gallouët, and Otared Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 5, 307–310 (French, with English summary). MR 734575
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
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
E. N. Dancer, Some mountain-pass solutions for small diffusion, Differential Integral Equations 16 (2003), no. 8, 1013–1024. MR 1989600
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
Manuel del Pino, Patricio L. Felmer, and Juncheng Wei, On the role of distance function in some singular perturbation problems, Comm. Partial Differential Equations 25 (2000), no. 1-2, 155–177. MR 1737546, DOI 10.1080/03605300008821511
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879, DOI 10.1007/BF01221125
R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 1-2, 53–72. MR 877892, DOI 10.1017/S0308210500019065
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
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
Louis Jeanjean and Kazunaga Tanaka, A remark on least energy solutions in $\textbf {R}^N$, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2399–2408. MR 1974637, DOI 10.1090/S0002-9939-02-06821-1
Pierre-Louis Lions, Symétrie et compacité dans les espaces de Sobolev, J. Functional Analysis 49 (1982), no. 3, 315–334 (French, with English summary). MR 683027, DOI 10.1016/0022-1236(82)90072-6
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283 (English, with French summary). MR 778974, DOI 10.1016/S0294-1449(16)30422-X
Yanyan Li and Louis Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1445–1490. MR 1639159, DOI 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q
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
Ezzat S. Noussair and Shusen Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc. (3) 76 (1998), no. 2, 427–452. MR 1490244, DOI 10.1112/S0024611598000148
Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original. MR 762825, DOI 10.1007/978-1-4612-5282-5
Michael Struwe, Variational methods, Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1078018, DOI 10.1007/978-3-662-02624-3
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