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Positive solutions for the p-Laplacian: application of the fibrering method

Published online by Cambridge University Press:  14 November 2011

Pavel Drábek  and
Stanislav I. Pohozaev
Pavel Drábek
Affiliation:
Department of Mathematics, University of West Bohemia, P.O. Box 314, 323 23 Plzeň, Czech Republic e-mail>: pdrabek@kma.zcu.cz
Stanislav I. Pohozaev
Affiliation:
Steklov Mathematical Institut, Russian Academy of Sciences, Vavilova 42, 117 966 Moscow, Russia e-mail: pohzaev@class.mi.ras.ru
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Abstract

Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 703 - 726
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1977).Google Scholar
2Allegretto, W. and Huang, Y.. Eigenvalues of the indefinite weight p-Laplacian in weighted ℝN spaces. Funkcial. Ekvac. (to appear).Google Scholar
3Anane, A.. Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I 305 (1987), 725–8.Google Scholar
4Barles, G.. Remarks on uniqueness results of the first eigenvalue of the p-Laplacian. Ann. Fac. Sci. Toulouse 9 (1988), 76–75.CrossRefGoogle Scholar
5Berestycki, H., Capuzzo-Dolcetta, I. and Nirenberg, L.. Problèmes elliptiques indéfinis et théorèmes de Liouville non linéaires. C. R. Acad. Sci. Paris Ser. I 317 (1993), 945–50.Google Scholar
6Drábek, P.. Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics 246 (Harlow: Longman, 1992).Google Scholar
7Drábek, P.. Nonlinear eigenvalue problem for the p-Laplacian in ℝN. Math. Nachr. 173 (1995), 131–9.CrossRefGoogle Scholar
8Drábek, P.. Strongly nonlinear degenerated and singular elliptic problems (to appear).Google Scholar
9Drábek, P. and Huang, Y.. Bifurcation problems for the p-Laplacian in ℝN. Trans. Amer. Math. Soc. (to appear).Google Scholar
10Drábek, P. and Huang, Y.. Multiple positive solutions of quasilinear elliptic equations in ℝN. Nonlinear. Anal. Theory, Meth. and Appl. (to appear).Google Scholar
11Fučik, S. and Kufner, A.. Nonlinear Differential Equations (Amsterdam: Elsevier, 1980).Google Scholar
12Lindqvist, P.. On the equation div (|∇ u|p–2 ∇u)+λ|u|p–2u= 0. Proc. Amer. Math. Soc. 109 (1990), 157–64.Google Scholar
13Pohozaev, S. I.. On eigenfunctions of quasilinear elliptic problems. Mat. Sb. 82 (1970), 192–212.Google Scholar
14Pohozaev, S. I.. On one approach to nonlinear equations. Dokl. Akad. Nauk 247 (1979), 1327–31 (in Russian).Google Scholar
15Pohozaev, S. I.. On a constructive method in the calculus of variations. Dokl. Akad. Nauk 298 (1988), 1330–3 (in Russian).Google Scholar
16Pohozaev, S. I.. On fibering method for the solution of nonlinear boundary value problems. Trudy Mat. Inst. Steklov. 192 (1990), 140–63 (in Russian).Google Scholar
17Serrin, J.. Local behavior of solutions of quasilinear equations. Ada Math. 111 (1964), 247–302.Google Scholar
18Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51 (1984), 126–50.CrossRefGoogle Scholar
19Trudinger, N. S.. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967), 721–47.CrossRefGoogle Scholar