Multiplicity of positive solutions for (p,q)-Laplace equations with two parameters

Bobkov, Vladimír; Tanaka, Mieko

doi:10.1142/s0219199721500085

Cited by 7 publications

(2 citation statements)

References 58 publications

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“…and denote by ̃︀ 𝐼 𝜇 𝜆 the corresponding truncated energy functional, i.e., (1.8) with Recently, the literature has been enriched with a few results on the local continuation of the branch of nonnegative solutions to various problems with respect to the parameter beyond a critical value of the type 𝜆 * characterizing the limit of applicability of the Nehari manifold method, see, e.g., [11,26,39,40]. To the best of our knowledge, the first contribution in this direction was made in [26] for the problem (𝑃 𝜆 ) in the superhomogeneous regime 𝑞 > 𝑝.…”

Section: Proof On P30mentioning

confidence: 99%

On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval

Bobkov¹,

Tanaka²

2021

Preprint

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We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation −Δ 𝑝 𝑢 = 𝜆|𝑢| 𝑝−2 𝑢 + 𝑎(𝑥)|𝑢| 𝑞−2 𝑢 in a bounded domain Ω ⊂ R 𝑁 , where 1 < 𝑞 < 𝑝, 𝜆 ∈ R, and 𝑎 is a continuous sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in {𝑥 ∈ Ω : 𝑎(𝑥) > 0}, when the parameter 𝜆 lies in a neighborhood of the criticalmain results, we show that if 𝑝 > 2𝑞 and either ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 = 0 or ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 > 0 is sufficiently small, then such solutions do exist in a right neighborhood of 𝜆 * . Here 𝜙 𝑝 is the first eigenfunction of the Dirichlet 𝑝-Laplacian in Ω. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case 𝑞 > 𝑝 or in the sublinear case 𝑞 < 𝑝 = 2 the nonexistence takes place for any 𝜆 ≥ 𝜆 * . Moreover, we prove that if 𝑝 > 2𝑞 and ∫︀ Ω 𝑎𝜙 𝑞 𝑝 𝑑𝑥 > 0 is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of 𝜆 * , two of which are strictly positive in {𝑥 ∈ Ω : 𝑎(𝑥) > 0}.

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Section: Proof On P30mentioning

confidence: 99%

On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval

Bobkov¹,

Tanaka²

2021

Preprint

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“…Different eigenvalue problems for the (p, q)-Laplacian have been extensively studied in recent years. In the context of Dirichlet boundary conditions we refer to the papers [4] by Bobkov and Tanaka,[8] by Colasuonno and Squassina, [14] by Marano and Mosconi, [15,16] by Marano, Mosconi and Papageorgiou, to the recent paper [20] due to Tanaka and finally to the references therein.…”

Section: Introductionmentioning

confidence: 99%

Multiple solutions for eigenvalue problems involving the (p,q)-Laplacian

Pucci

2023

Stud. Univ. Babes-Bolyai Math.

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"This paper is devoted to a subject that Professor Csaba Varga suggested during his frequent visits to the University of Perugia and in my regular stays at the ""Babe s-Bolyai"" University. More speci cally, continuing the work started in [7] jointly with Professor Varga, here we establish the existence of two nontrivial (weak) solutions of some one parameter eigenvalue (p,q)-Laplacian problems under homogeneous Dirichlet boundary conditions in bounded domains of R^N."

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