L ∞-Estimates for nonlinear elliptic Neumann boundary value problems

Winkert, Patrick

doi:10.1007/s00030-009-0054-5

Cited by 49 publications

(38 citation statements)

References 14 publications

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“…As in the proof of Proposition 3.3, using the nonlinear regularity theory (see [14], [27] and [15]) and the nonlinear maximum principle (see [25]), we have u ∈ int C + . So, we can find C 9 > 0 such that u(z) ≥ C 9 for all z ∈ Ω. Let…”

Section: Corrected Proofmentioning

confidence: 96%

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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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Section: Corrected Proofmentioning

confidence: 96%

“…We have Motreanu and Papageorgiou [20]). From Hu and Papageorgiou [14] and Winkert [27], we have that u ∈ L ∞ (Ω). So, we can apply the regularity result of Lieberman [15] (p. 320) and infer that u ∈ C + \ {0}.…”

Section: Positive Solutionsmentioning

confidence: 99%

Nonlinear, nonhomogeneous parametric Neumann problems

Aizicovici¹,

Papageorgiou²,

Staicu³

2016

TMNA

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“…Since r > p, from (17) it follows that ψ is coercive. Also, it is sequentially weakly lower semicontinuous.…”

Section: Propositionmentioning

confidence: 98%

“…From Winkert [17], we have thatû λ ∈ L ∞ (Ω). So, we can apply Theorem 2 of Lieberman [13] and obtain thatû λ ∈ C + \ {0}.…”

Section: Integrating By Parts We Havementioning

confidence: 99%

Positive solutions for perturbations of the eigenvalue problem for the Robin p-Laplacian

Papageorgiou

Rădulescu

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study perturbations of the eigenvalue problem for the Robin p-Laplacian. First we consider the case of a (p − 1)-sublinear perturbation and prove existence, nonexistence and uniqueness of positive solutions. Then we deal with the case of a (p − 1)-superlinear perturbation which need not satisfy the Ambrosetti-Rabinowitz condition and prove a multiplicity result for positive solutions. Our approach uses variational methods together with suitable truncation and perturbation techniques.

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“…We have ). From Hu-Papageorgiou [14] and Winkert [27], we have that u ∈ L ∞ (Ω) . So, we can apply the regularity result of Lieberman [15] (p. 320) and infer that u ∈ C + \ {0} .…”

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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L ∞-Estimates for nonlinear elliptic Neumann boundary value problems

Cited by 49 publications

References 14 publications

Nonlinear, nonhomogeneous parametric Neumann problems

Nonlinear, nonhomogeneous parametric Neumann problems

Positive solutions for perturbations of the eigenvalue problem for the Robin p-Laplacian

Nodal solutions for nonlinear nonhomogeneous Neumann equations

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