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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Multiplicity of solutions of the Dirichlet problem for an equation with the $ p$-Laplacian in a three-dimensional spherical layer

Author: S. B. Kolonitskiĭ
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 485-495
MSC (2010): Primary 35J92
Published electronically: March 18, 2011
MathSciNet review: 2729947
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Abstract: The equation $ -\Delta_p u = u^{q-1}$ with zero Dirichlet condition on the boundary is considered in a three-dimensional spherical layer. The existence of arbitrarily many distinct positive solutions in a sufficiently thin layer is proved.

References [Enhancements On Off] (What's this?)

  • 1. A. I. Nazarov, On solution to the Dirichlet problem for an equation with $ p$-Laplacian in a spherical layer, Trudy S.-Peterburg. Mat. Obshch. 10 (2004), 33-62; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 214, Amer. Math. Soc., Providence, RI, 2005, pp. 29-57. MR 2181511 (2006i:35101)
  • 2. Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differential Equations 83 (1990), 348-367. MR 1033192 (91a:35073)
  • 3. C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), 429-437. MR 0760381 (86e:35055)
  • 4. A. I. Nazarov, The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers, Probl. Mat. Anal., vol. 20, Nauchn. Kniga, Novosibirsk, 2000, pp. 171-190; English transl., J. Math. Sci. (N.Y.) 102 (2000), no. 5, 4473-4486. MR 1807067 (2001k:35065)
  • 5. N. Mizoguchi and T. Suzuki, Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math. 22 (1996), 199-215. MR 1434392 (97j:35052)
  • 6. J. Byeon, Existence of many nonequivalent nonradial positive solutions of semilinear elliptic equations on three-dimensional annuli, J. Differential Equations 136 (1997), 136-165. MR 1443327 (98b:35060)
  • 7. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, 2nd ed., Nauka, Moscow, 1973; English transl. of 1st ed., Acad. Press, New York-London, 1968. MR 0509265 (58:23009); MR 0244627 (39:5941)
  • 8. R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), 19-30. MR 0547524 (81c:58026)
  • 9. N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. MR 0226198 (37:1788)

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Additional Information

S. B. Kolonitskiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, 28 Universitetskii Prospekt, Peterhoff, St. Petersburg 198504, Russia

Keywords: $p$-Laplacian, existence of many solutions
Received by editor(s): September 22, 2009
Published electronically: March 18, 2011
Additional Notes: Supported by RFBR (grant no. 08-01-00748) and by the grant NSh–227.2008.1 for support of leading scientific schools
Article copyright: © Copyright 2011 American Mathematical Society

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