Nontrivial Solutions for<i>p</i>-Laplace Equations with Right-Hand Side Having<i>p</i>-Linear Growth at Infinity

Cingolani, Silvia; Degiovanni, Marco

doi:10.1080/03605300500257594

Cited by 105 publications

(89 citation statements)

References 14 publications

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“…Such operators are important in quantum physics, see, for example, Benci-FortunatoPisani [8]. Recently, there have been papers dealing with the (p, q) −Laplacian, see Cingolani-Degiovanni [12], Figueiredo [18] and Medeiros-Perera [30].…”

Section: Preliminary Results It What Followsmentioning

confidence: 99%

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

Aizicovici

Papageorgiou

Staicu

2015

Methods and Applications of Analysis

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Abstract. We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is "concave" (i.e., (p − 1) − sublinear) near zero and "convex" (i.e., (p − 1) − superlinear) near ±∞. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter λ > 0, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case (p = 2), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign.

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Section: Preliminary Results It What Followsmentioning

confidence: 99%

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

Aizicovici

Papageorgiou

Staicu

2015

Methods and Applications of Analysis

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“…Recently there have been existence and multiplicity results for equations driven by such operators. We mention the works of Aizicovici, Papageorgiou and Staicu [4], Cingolani and Degiovanni [7], Mugnai and Papageorgiou [22], Papageorgiou and Radulescu [23], Sun [26].…”

Section: Corrected Proofmentioning

confidence: 99%

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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“…Recently there have been existence and multiplicity results for equations driven by such operators. We mention the works of AizicoviciPapageorgiou-Staicu [4], Cingolani-Degiovanni [7], Mugnai-Papageorgiou [22], Papa-georgiou-Radulescu [23], Sun [26].…”

Section: Examplesmentioning

confidence: 99%

Nodal solutions for nonlinear nonhomogeneous Neumann equations

Aizicovici¹,

Papageorgiou²,

Staicu³

2016

TMNA

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Abstract. We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator and with a Caratheodory reaction f (t, x) which is p−superlinear in x without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of the positive solutions on the parameter λ > 0, we show the existence of a smallest positive solution u λ and investigate the properties of the map λ → u λ . Finally we also show the existence of nodal solutions.

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Nontrivial Solutions forp-Laplace Equations with Right-Hand Side Havingp-Linear Growth at Infinity

Cited by 105 publications

References 14 publications

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

Constant sign and nodal solutions for nonlinear elliptic equations with combined nonlinearities

Nonlinear, nonhomogeneous parametric Neumann problems

Nodal solutions for nonlinear nonhomogeneous Neumann equations

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