A multiplicity theorem for hemivariational inequalities with a p -Laplacian-like differential operator

Papageorgiou, Nikolaos S.; Rocha, Eugénio M.; Staicu, Vasile

doi:10.1016/j.na.2007.06.023

Cited by 32 publications

(10 citation statements)

References 10 publications

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“…Problems driven by nonhomogeneous differential operators were investigated by Dabuleanu and Radulescu [9], De Napoli and Mariani [11], Duc and Vu [12], Hu and Papageorgiou [18], Kourogenis and Papageorgiou [21], Kristaly et al [22], Kyritsi et al [28] and Papageorgiou et al [29].…”

Section: Introductionmentioning

confidence: 99%

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On p-superlinear equations with a nonhomogeneous differential operator

Aizicovici

Papageorgiou

Staicu

2012

Nonlinear Differ. Equ. Appl.

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Section: Introductionmentioning

confidence: 99%

“…Multiplicity results were proved in [11] (under symmetry conditions on the reaction term) and in [22,23,28,29] for certain classes of parametric equations. The p-superlinear case is considered only in [28] under a different hypothesis on the reaction.…”

Section: Introductionmentioning

confidence: 99%

On p-superlinear equations with a nonhomogeneous differential operator

Aizicovici

Papageorgiou

Staicu

2012

Nonlinear Differ. Equ. Appl.

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“…Theorem 4.2 improves substantially Theorem 3.5 in [18]. corrected proof 24 S. Aizicovici -N. S. Papageorgiou -V. Staicu…”

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confidence: 74%

Nonlinear, nonhomogeneous parametric Neumann problems

Aizicovici¹,

Papageorgiou²,

Staicu³

2016

TMNA

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Abstract. We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction f which is p-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of positive solutions on the parameter λ > 0, show the existence of a smallest positive solution u λ and investigate properties of the map λ → u λ . Finally, we show the existence of nodal solutions. IntroductionIn this paper we study the following nonlinear parametric Neumann problem: Here Ω ⊂ R N is a bounded domain with a C 2 -boundary ∂Ω.The map a : R N → R N is continuous, strictly monotone and satisfies certain regularity conditions which are listed in hypotheses H(a) (see Section 2). These hypotheses are general enough to incorporate in our framework many differential operators of interest, such as the p-Laplacian. Also λ > 0 is a parameter and f is a Carathéodory function (i.e. for all x ∈ R, z → f (z, x) is measurable and for almost all z ∈ Ω, x → f (z, x) is continuous) which exhibits a (p − 1)-superlinear growth in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition (AR-condition for short). Our work is motivated by a recent paper of Motreanu, Motreanu and Papageorgiou [18], who produced constant sign and nodal solutions. Our results complement and improve those of [18]. More precisely, the authors in [18] produced positive solutions for problem (P λ ) but did not give the precise dependence of the set of positive solutions on the parameter λ > 0. Here, we prove a bifurcation-type theorem for large values of λ, which gives a complete picture of the set of positive solutions as the parameter varies. Moreover, in [18] nodal (that is, sign-changing) solutions were produced only for the particular case of equations driven by the p-Laplacian. In contrast, here we generate nodal solutions for the general case. We stress that the p-Laplacian differential operator is homogeneous, while the differential operator in (P λ ) is not. Hence, the methods and techniques used in [18] fail in the present setting, and so a new approach is needed. Finally, we mention that a bifurcation near infinity for a different class of p-Laplacian Dirichlet problems was recently produced by Gasinski and Papageorgiou [12].In the next section, we review the main mathematical tools which will be used in this paper. We also present the hypotheses on the map y → a(y) and state some useful consequences of them.

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“…In what follows, for notational economy, we write . We introduce the nonlinear map A : W → W ∗ , defined by From Papageorgiou-Rocha-Staicu [ 21 ], we have the following result concerning the map A .…”

Section: Mathematical Background - Hypothesesmentioning

confidence: 99%

Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction

Gasiński

Papageorgiou

2014

J Dyn Control Syst

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We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and comparison techniques, we prove two multiplicity theorems providing sign information for all the solutions.

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A multiplicity theorem for hemivariational inequalities with a p -Laplacian-like differential operator

Cited by 32 publications

References 10 publications

On p-superlinear equations with a nonhomogeneous differential operator

On p-superlinear equations with a nonhomogeneous differential operator

Nonlinear, nonhomogeneous parametric Neumann problems

Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction

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