A Picone's identity for the p-Laplacian and applications

Allegretto, W.; Huang, Yin Xi

doi:10.1016/s0362-546x(97)00530-0

Cited by 356 publications

(314 citation statements)

References 11 publications

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“…Moreover, as in the proof of Proposition 9, using the nonlinear Picone's identity (see [2]), we have: Proposition 17. If hypotheses H(β) and H(f ) 1 hold, thenλ 1 ∈ P and so P = (−∞,λ 1 ).…”

Section: Proposition 16mentioning

confidence: 86%

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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Section: Proposition 16mentioning

confidence: 86%

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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“…The proof is based on the somehow weird inequality (8), which is in turn a consequence of the convexity of the function |ξ| p , p ∈ (1, +∞), with respect to the variable ξ ∈ R N . Several related results are found in the literature: in particular, Díaz-Saá's inequality [5,10] and Picone's identity [2]. An elegant uniqueness result exploiting the strict convexity of the associated functional H p (u) with respect to the function u p (hidden convexity) is found in [4].…”

Section: Theorem 11 (Weak Comparison Principle)mentioning

confidence: 91%

Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian

Greco¹

2014

Publicacions Matemàtiques

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A comparison principle for the subdiffusive p-Laplacian in a possibly nonsmooth and unbounded open set is proved. The result requires that the involved sub and supersolution are positive, and the ratio of the former to the latter is bounded. As an application, constrained radial symmetry for overdetermined problems is obtained. More precisely, both Dirichlet and Neumann conditions are prescribed on the boundary of a bounded open set, and the Neumann condition depends on the distance from the origin. The domain of the problem, unknown at the beginning, turns out to be a ball centered at the origin if a positive solution exists. Counterexamples are also discussed.

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“…In fact, Picone's identity in [1] is proved for C 1 functions ξ 1 , ξ 2 , however, one can show that Picone's identity remains true if…”

Section: Generalized Principal Eigenvaluementioning

confidence: 99%

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

Nguyen

2015

Journal of Functional Analysis

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A Picone's identity for the p-Laplacian and applications

Cited by 356 publications

References 11 publications

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

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

Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

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