On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problemwhere Ω is a bounded domain in R N , g ∈ C(Ω × R, R), and the function φ(s) = sa(|s|) is an increasing homeomorphism from R onto R. Under appropriate conditions on φ, g, and the Orlicz-Sobolev conjugate Φ * of Φ(s) = s 0 φ(t) dt, (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type. Mathematics Subject Classification (1991):35J20, 35J25

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“…On the other hand, it follows from [7] that if Φ satisfies a global ∆ 2 condition then there exists a best positive constant λ 1 such that…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Mountain pass type solutions for quasilinear elliptic equations

Clément

Garcı́a-Huidobro

Manásevich

et al. 2000

Calculus of Variations and Partial Differential Equations

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113

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“…Relations (2) and (7) enable us to apply Theorem 2.2 in [12] (see also Theorem 8.33 in [3]) in order to obtain that…”

Section: Remarkmentioning

confidence: 99%

A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

Mihăilescu¹,

Rădulescu²

2009

MATH. SCAND.

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Abstract. We study the nonlinear eigenvalue problem −div(a(|∇u|)∇u) = λ|u| q(x)−2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary, q is a continuous function, and a is a nonhomogeneous potential. We establish sufficient conditions on a and q such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases a(t) = t p−2 log(1 + t r ) and a(t) = t p−2 [log(1 + t)] −1 .

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“…Under the assumptions (φ 1 )-(φ 3 ), by similar arguments as [21], we can prove that I 1 is well-defined and of class C 1 (W, R) and…”

Section: Existencementioning

confidence: 66%

Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

Wang¹,

Zhang²,

Fang³

2017

J. Nonlinear Sci. Appl.

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In this paper, we investigate the following nonlinear and non-homogeneous elliptic systemwhere Ω is a bounded domain in R N (N 2) with smooth boundary ∂Ω, functions φ i (t)t (i = 1, 2) are increasing homeomorphisms from R + onto R + . When F satisfies some (φ 1 , φ 2 )-superlinear and subcritical growth conditions at infinity, by using the mountain pass theorem we obtain that system has a nontrivial solution, and when F satisfies an additional symmetric condition, by using the symmetric mountain pass theorem, we obtain that system has infinitely many solutions. Some of our results extend and improve those corresponding results in Carvalho et al. [M. L. M. Carvalho,

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On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting

Cited by 136 publications

References 22 publications

Mountain pass type solutions for quasilinear elliptic equations

Mountain pass type solutions for quasilinear elliptic equations

A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

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