Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T22:50:43.997Z Has data issue: false hasContentIssue false
  • English
  • Français

A result of multiplicity of solutions for a class of quasilinear equations

Part of: Close-to-elliptic equations and systems Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  23 February 2012

Claudianor O. Alves ,
Giovany M. Figueiredo  and
Uberlandio B. Severo
Claudianor O. Alves
Affiliation:
Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande PB, Brazil ([email protected])
Giovany M. Figueiredo
Affiliation:
Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande PB, Brazil ([email protected]) Faculdade de Matemática, Universidade Federal do Pará, 66075-110 Belém PA, Brazil ([email protected])
Uberlandio B. Severo
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brazil ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem −Lpu + |u|p−2u = h(u) in Ωλ, u = 0 on ∂Ωλ, where Ωλ = λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter, Lpu ≐ Δpu + Δp(u2)u and the nonlinear term h(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

Keywords

variational methodsquasilinear equationsstanding wave solutionsLyusternik–Schnirelmann category

MSC classification

Secondary: 35A15: Variational methods 35H30: Quasi-elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 291 - 309
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Alves, C. O., Existence and multiplicity of solution for a class of quasilinear equations, Adv. Nonlin. Studies 5 (2005), 73–87.CrossRefGoogle Scholar
2.Alves, C. O., Figueiredo, G. M. and Severo, U. B., Multiplicity of positive solutions for a class of quasilinear problems, Adv. Diff. Eqns 14 (2009), 911–942.Google Scholar
3.Benci, V. and Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Ration. Mech. Analysis 114 (1991), 79–83.CrossRefGoogle Scholar
4.Benci, V. and Cerami, G., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. PDEs 02 (1994), 29–48.CrossRefGoogle Scholar
5.Berestycki, H. and Lions, P. L., Nonlinear scalar field equations, I, Existence of a ground state, Arch. Ration. Mech. Analysis 82 (1983), 313–346.CrossRefGoogle Scholar
6.Borovskii, A. and Galkin, A., Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP 77 (1983), 562–573.Google Scholar
7.Brizhik, L., Eremko, A., Piette, B. and Zakrzewski, W. J., Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity 16 (2003), 1481–1497.CrossRefGoogle Scholar
8.Colin, M. and Jeanjean, L., Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlin. Analysis 56 (2004), 213–226.CrossRefGoogle Scholar
9.do Ó, J. M. and Severo, U. B., Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Analysis 8 (2009), 621–644.CrossRefGoogle Scholar
10.do Ó, J. M., Miyagaki, O. and Soares, S., Soliton solutions for quasilinear Schrödinger equations: the critical exponential case, Nonlin. Analysis 67 (2007), 3357–3372.CrossRefGoogle Scholar
11.Floer, A. and Weinstein, A., Nonspreading wave packets for the packets for the cubic Schrodinger with a bounded potential, J. Funct. Analysis 69 (1986), 397–408.CrossRefGoogle Scholar
12.Hartmann, B. and Zakrzewski, W. J., Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B68 (2003), 184302.CrossRefGoogle Scholar
13.Jeanjean, L. and Tanaka, K., A positive solution for a nonlinear Schrödinger equation on ℝN, Indiana Univ. Math. J. 54 (2005), 443–464.CrossRefGoogle Scholar
14.Kosevich, A. M., Ivanov, B. A. and Kovalev, A. S., Magnetic solitons in superfluid films, J. Phys. Soc. Jpn 50 (1981), 3262–3267.Google Scholar
15.Kurihura, S., Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Jpn 50 (1981), 3262–3267.CrossRefGoogle Scholar
16.Liu, J. and Wang, Z. Q., Soliton solutions for quasilinear Schrödinger equations, I, Proc. Am. Math. Soc. 131 (2002), 441–448.CrossRefGoogle Scholar
17.Liu, J., Wang, Y. and Wang, Z. Q., Soliton solutions for quasilinear Schrödinger equations, II, J. Diff. Eqns 187 (2003), 473–493.CrossRefGoogle Scholar
18.Liu, J., Wang, Y. and Wang, Z. Q., Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. PDEs 29 (2004), 879–901.CrossRefGoogle Scholar
19.Makhankov, V. G. and Fedyanin, V. K., Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104 (1984), 1–86.CrossRefGoogle Scholar
20.Poppenberg, M., Schmitt, K. and Wang, Z. Q., On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. PDEs 14 (2002), 329–344.CrossRefGoogle Scholar
21.Ritchie, B., Relativistic self-focusing and channel formation in laser–plasma interactions, Phys. Rev. E50 (1994), 687–689.Google ScholarPubMed
22.Severo, U. B., Estudo de uma classe de equações de Schrödinger quase-lineares, Doctoral Dissertation, Unicamp (2007).Google Scholar
23.Severo, U. B., Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian, Electron. J. Diff. Eqns 2008(56) (2008), 1–16.Google Scholar
24.Takeno, S. and Homma, S., Classical planar Heinsenberg ferromagnet, complex scalar fields and nonlinear excitations, Prog. Theor. Phys. 65 (1981), 172–189.CrossRefGoogle Scholar
25.Trudinger, N. S., On Harnack type inequalities and their applications to quasilinear elliptic equations, Commun. Pure Appl. Math. 20 (1967), 721–747.CrossRefGoogle Scholar
26.Vásquez, J. L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.CrossRefGoogle Scholar
27.Willem, M., Minimax theorems (Birkhäuser, 1996).CrossRefGoogle Scholar