A multiplicity result for the p-Laplacian involving a parameter

Friedemann Brock, Leonelo Iturriaga, Pedro Ubilla

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38 Citations (Scopus)


We study existence and multiplicity of positive solutions for the following problem {-Δpu = λ f(x,u) in Ω u = 0 on ∂Ω' where λ is a positive parameter, Ω is a bounded and smooth domain in ℝN, p ∈ (1, N), f(x,t) behaves, for instance, like o(|t|p-1) near 0 and +∞, and satisfies some further properties. In particular, our assumptions allow us to consider both positive and sign changing nonlinearitites f, the latter describing logistic as well as reaction-diffusion processes. By using sub- and supersolutions and variational arguments, we prove that there exists a positive constant λ such that the above problem has at least two positive solutions for λ > λ, at least one positive solution for λ = λ and no solution for λ < λ. An important rôle plays the fact that local minimizers of certain functionals in the C1-topology are also minimizers in W01,p(Ω). We give a short new proof of this known result. © 2008 Birkhaueser.
Original languageEnglish
Pages (from-to)1371-1386
Number of pages16
JournalAnnales Henri Poincare
Publication statusPublished - 1 Nov 2008
Externally publishedYes

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