Principal Eigenvalues and Sturm Comparison via Picone's Identity

Allegretto, W.; Huang, Yin Xi

doi:10.1006/jdeq.1998.3596

Cited by 38 publications

(77 citation statements)

References 8 publications

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“…The next section gives preliminary results concerning positive solutions of the equation Q ′ (u) = 0. In particular, it introduces the generalized Picone's identity [2,3] which is a crucial tool in our study. Section 3 is devoted to the proof of Theorem 1.6.…”

Section: Introductionmentioning

confidence: 99%

Ground state alternative for p-Laplacian with potential term

2006

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Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q ′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

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Section: Introductionmentioning

confidence: 99%

Ground state alternative for p-Laplacian with potential term

2006

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“…Our criteria for positive µ depend on the following lemmas, which extend to the present situation some of the arguments of [3], [4], [11]: Proof. We simply note by direct calculation:…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…It involves the introduction of an arbitrary vector field b , which then must be suitably chosen. This approach is related to Picone Identity arguments (see [3], [4], [6] and elsewhere) if p is a constant, since the "best" fields b usually involve at least in part the gradient of the eigenfunction associated with the least eigenvalue. But in the present context, where terms involving ∇p arise, the former procedure appears more advantageous.…”

Section: Introductionmentioning

confidence: 99%

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Form estimates for the 𝑝(𝑥)-Laplacean

Allegretto

2007

Proc. Amer. Math. Soc.

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Abstract. We consider the problem of establishing conditions on p(x) that ensure that the form associated with the p(x)-Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that -unlike the p = constant case -this is not possible if p has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for p = 2, involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in Ω ⊂ R n with n ≥ 1, and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous p(x). Our basic criteria involve restrictions on p(x) and its gradient.

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“…We refer to Kreith [21,22], Swanson [29] for Picone identities and Sturmian comparison theorems for linear elliptic equations, and to Allegretto [3], Allegretto and Huang [5,6], Bognár and Došlý [9], Dunninger [13], Jaroš et al [16,17,19,20], Kusano et al [23], Yoshida [36,38], Tiryaki [31], Tiryaki and Sahiner [33] for Picone identities, Sturmian comparison and/or oscillation theorems for half linear elliptic equations.…”

Section: Introductionmentioning

confidence: 99%

Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities

Tı̇ryakı̇¹,

Sahiner²,

Mısırlı³

2016

Proceedings of the 10'th Colloquium on the Qualitative Theory of Differential Equations (July 1–4, 2015, Szeged, Hungary) Edite

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Abstract. Sturm theorems have appeared as one of the fundamental subjects in qualitative theory to determine properties of the solutions of differential equations. Motivated by some recent developments for half-linear type elliptic equations, we obtain Piconetype inequalities for two pairs of elliptic type equations with damping and external terms in order to establish Sturmian comparison theorems. Some oscillation results are given as applications.

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Principal Eigenvalues and Sturm Comparison via Picone's Identity

Cited by 38 publications

References 8 publications

Ground state alternative for p-Laplacian with potential term

Ground state alternative for p-Laplacian with potential term

Form estimates for the 𝑝(𝑥)-Laplacean

Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities

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