The smallest characteristic numbers in a certain exceptional case

Bôcher, Maxime

doi:10.1090/s0002-9904-1914-02560-1

Cited by 37 publications

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“…A basic result in the elementary theory of linear partial differential equations asserts that the spectrum of the Laplace operator in H was considered by Bocher [5], Hess and Kato [16], Minakshisundaram and Pleijel [20,22]. For instance, Minakshisundaram and Pleijel proved that the above eigenvalue problem has an unbounded sequence of positive eigenvalues if a ∈ L ∞ (Ω), a ≥ 0 in Ω, and a > 0 in Ω 0 ⊂ Ω, where |Ω 0 | > 0.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Mihăilescu

Rădulescu

2007

Proc. Amer. Math. Soc.

204

119

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Abstract. We consider the nonlinear eigenvalue problemin Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary and p, q are continuous functions on Ω such that 1 < inf Ω q < inf Ω p < sup Ω q, sup Ω p < N, and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle. Introduction and preliminary resultsA basic result in the elementary theory of linear partial differential equations asserts that the spectrum of the Laplace operator in H In this paper we are concerned with the nonhomogeneous eigenvalue problemwhere Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, λ > 0 is a real number, and p, q are continuous on Ω.The case p(x) = q(x) was considered by Fan, Zhang and Zhao in [14] who, using the Ljusternik-Schnirelmann critical point theory, established the existence of a sequence of eigenvalues. Denoting by Λ the set of all nonnegative eigenvalues, Fan, Zhang and Zhao showed that sup Λ = +∞, and they pointed out that only under additional assumptions we have inf Λ > 0. We remark that for the p-Laplace operator (corresponding to p(x) ≡ p) we always have inf Λ > 0.In this paper we study problem (1.1) under the basic assumptionOur main result establishes the existence of a continuous family of eigenvalues for problem (1.1) in a neighborhood of the origin. More precisely, we show that there exists λ > 0 such that any λ ∈ (0, λ ) is an eigenvalue for problem (1.1). We start with some preliminary basic results on the theory of Lebesgue-Sobolev spaces with variable exponent. For more details we refer to the book by Musielak [21] and the papers by Edmunds et al. [8,9,10], Kovacik and Rákosník [17], and Samko and Vakulov [24].Assume that p ∈ C(Ω) andFor any h ∈ C + (Ω) we defineFor any p(x) ∈ C + (Ω), we define the variable exponent Lebesgue spaceu is a measurable real-valued function andWe define a norm, the so-called Luxemburg norm, on this space by the formulaLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON A NONHOMOGENEOUS QUASILINEAR EIGENVALUE PROBLEM 2931We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces. If 0 < |Ω| < ∞ and p 1 , p 2 are variable exponent, so that p 1 (x) ≤ p 2 (x) almost everywhere in Ω, then there exists the continuous embeddingWe denote by(Ω) the Hölder type inequalityholds true. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ p(x) :, then the following relations hold true: (Ω), · ) is a separable and reflexive Banach space. We note that if s(x) ∈ C + (Ω) and s(x) < p (x) for all x ∈ Ω, then the embedding W(Ω) is compact and continuous, whereWe refer to Kováčik and Rákosník [17] for more properties of Lebesgue and Sobolev spaces with variable exponen...

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Mihăilescu

Rădulescu

2007

Proc. Amer. Math. Soc.

204

119

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“…The study of the linear ordinary differential equation case, however, goes back to Bocher [3]. Attention has been confined mainly to the cases of Dirichlet (α = ∞) and Neumann boundary conditions.…”

Section: −∆U(x) = λG(x)u(x)mentioning

confidence: 99%

Boundedness and monotonicity of principal eigenvalues for boundary value problems with indefinite weight functions

Afrouzi

2002

International Journal of Mathematics and Mathematical Sciences

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We study the principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem:where ∆ is the standard Laplace operator, D is a bounded domain with smooth boundary, g : D → R is a smooth function which changes sign on D and α ∈ R. We discuss the relation between α and the principal eigenvalues.2000 Mathematics Subject Classification: 35J15, 35J25.

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“…The ordinary differential equation versions of (1.3) were studied by Picone [7] and Bocher [2]. Motivated by Fleming's paper, Brown and Lin [3], and Hess and Kato [5] studied the eigenvalues and eigenfunctions of (1.3) in the partial differential equation case.…”

Section: −∆U(x) = λG(x)u(x) X ∈ B R (0);mentioning

confidence: 99%

On the continuity of principal eigenvalues for boundary valueproblems with indefinite weight function with respect to radiusof balls in ℝ^N

Afrouzi

2002

International Journal of Mathematics and Mathematical Sciences

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The smallest characteristic numbers in a certain exceptional case

Cited by 37 publications

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On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Boundedness and monotonicity of principal eigenvalues for boundary value problems with indefinite weight functions

On the continuity of principal eigenvalues for boundary valueproblems with indefinite weight function with respect to radiusof balls in ℝ^N

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The smallest characteristic numbers in a certain exceptional case

Cited by 37 publications

References 0 publications

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

Boundedness and monotonicity of principal eigenvalues for boundary value problems with indefinite weight functions

On the continuity of principal eigenvalues for boundary valueproblems with indefinite weight function with respect to radiusof balls in ℝN

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On the continuity of principal eigenvalues for boundary valueproblems with indefinite weight function with respect to radiusof balls in ℝ^N