Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

Samko, Stefan; Vakulov, B. G.

doi:10.1016/j.jmaa.2005.02.002

Cited by 59 publications

(32 citation statements)

References 16 publications

(15 reference statements)

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“…For more details we refer to the book by Musielak [27] and the papers by Edmunds et al [8,9,10], Kováčik and Rákosník [17], Mihȃilescu and Rȃdulescu [21], and Samko and Vakulov [29].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

Mihăilescu

Rădulescu

2010

JAMA

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In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the particular eigenvalue given by the infimum of the associated Rayleigh quotient when the variable potential lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

Mihăilescu

Rădulescu

2010

JAMA

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“…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 …”

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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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“…The boundedness of the terms A ++ and A −− was shown in [8] without condition (1.11). So we only have to treat the terms A +− and A −+ .…”

Section: The Case Of the Spatial Potential Operatormentioning

confidence: 94%

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

Shargorodsky

Vakulov

2007

Journal of Mathematical Analysis and Applications

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In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p(·) → q(·)-theorems were proved for the Riesz potential operator I α in the weighted Lebesgue generalized spaces L p(·) (R n , ρ) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x 0 and to infinity, under an additional condition relating the weight exponents at x 0 and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L p(·) (S n , ρ) on the unit sphere S n in R n+1 are also improved in the same way.

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Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators

Cited by 59 publications

References 16 publications

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

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

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

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