Wang’s multiplicity result for superlinear $(p,q)$–equations without the Ambrosetti–Rabinowitz condition

Mugnai, Dimitri; Papageorgiou, Nikolaos S.

doi:10.1090/s0002-9947-2013-06124-7

Cited by 119 publications

(94 citation statements)

References 25 publications

(12 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We mention that similar or different extensions of the AR-superlinearity condition can be found in Aizicovici, Papageorgiou and Staicu [3], Costa and Magalhães [8], Li and Yang [16], and Mugnai and Papageorgiou [22].…”

Section: Positive Solutionsmentioning

confidence: 55%

See 1 more Smart Citation

Nonlinear, nonhomogeneous parametric Neumann problems

Aizicovici¹,

Papageorgiou²,

Staicu³

2016

TMNA

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Positive Solutionsmentioning

confidence: 55%

“…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

Self Cite

View full text Add to dashboard Cite

show abstract

“…We mention that similar or different extensions of the AR-superlinearity condition can be found in Aizicovici-Papageorgiou-Staicu [3], Costa-Magalhães [8], Li-Yang [16], and Mugnai-Papageorgiou [22].…”

Section: Positive Solutionsmentioning

confidence: 69%

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Recently there have been some existence and multiplicity results for such equations. In this direction, we mention the works of Aizicovici et al [2], Barile and Figueiredo [5], Cingolani and Degiovanni [9], Gasiński and Papageorgiou [17,20], Gasiński et al [21], Mugnai and Papageorgiou [28], Papageorgiou and Rǎdulescu [31,32], Papageorgiou and Smyrlis [33], Sun [37] and Sun et al [38]. Of the aforementioned works, only Papageorgiou and Rǎdulescu [31] and Gasiński and Papageorgiou [20] deal with problems having an asymmetric reaction term.…”

Section: Introductionmentioning

confidence: 99%

Asymmetric (p, 2)-equations with double resonance

Gasiński

Papageorgiou

2017

Calc. Var.

Self Cite

View full text Add to dashboard Cite

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a pLaplacian and a Laplacian [a ( p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions. Mathematics Subject Classification

show abstract

Wang’s multiplicity result for superlinear $(p,q)$–equations without the Ambrosetti–Rabinowitz condition

Cited by 119 publications

References 25 publications

Nonlinear, nonhomogeneous parametric Neumann problems

Nonlinear, nonhomogeneous parametric Neumann problems

Nodal solutions for nonlinear nonhomogeneous Neumann equations

Asymmetric (p, 2)-equations with double resonance

Contact Info

Product

Resources

About