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Existence and multiplicity of solutions of quasilinear equations with convex or nonconvex reaction term

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Abstract

We give existence, nonexistence and multiplicity results of nonnegative solutions for Dirichlet problems of the form

$$ - {\Delta_p}v = \lambda f(x){\left( {1 + g(v)} \right)^{p - 1}}\quad {\text{in}}\ \Omega,\quad u = 0\quad {\text{on}}\ \partial \Omega, $$

where Δ p is the p-Laplacian (p > 1), g is nondecreasing, superlinear, and possibly convex, λ > 0, and f ∈ L 1 (Ω), f ≥ 0. New information on the extremal solutions is given. Equations with measure data are also considered.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200, Tours, France

    H. A. Hamid & M. F. Bidaut-Veron

Authors
  1. H. A. Hamid
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  2. M. F. Bidaut-Veron
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Correspondence to H. A. Hamid.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 35, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations. Part 1, 2010.

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Hamid, H.A., Bidaut-Veron, M.F. Existence and multiplicity of solutions of quasilinear equations with convex or nonconvex reaction term. J Math Sci 170, 324–331 (2010). https://doi.org/10.1007/s10958-010-0088-6

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  • DOI: https://doi.org/10.1007/s10958-010-0088-6

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