An Approximation Result in Generalized Anisotropic Sobolev Spaces and Applications

We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic \vec p-Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.

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“…The space W , p(•) (Ω), • is a separable and re exive Banach space (cf. [6,24]). Throughout the rest of this paper, we assume p > N.…”

Section: Preliminariesmentioning

confidence: 99%

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

Böhner

Caristi

Gharehgazlouei

et al. 2020

Nonautonomous Dynamical Systems

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“…, p N (•)) and 1 p i (x) + 1 p i (x) = 1 (cf. [5] for the constant exponent case). For each F ∈ W −1, p (•) (Ω) there exists F 0 ∈ (L p + (Ω)) and…”

Section: Preliminarymentioning

confidence: 99%

Existence of Renormalized Solutions for Some Anisotropic Quasilinear Elliptic Equations

Ahmedatt

Aberqi

Hjiaj

et al. 2020

KgJMat

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In this paper, we consider a class of anisotropic quasilinear elliptic equations of the type ( | ∑N { − ∂ia (x, u, ∇u ) + |u|s(x )− 1u = f (x,u ), in Ω, i |( i=1 u = 0 on ∂ Ω, where f(x,s) is a Carathéodory function which satisﬁes some growth condition. We prove the existence of renormalized solutions for our Dirichlet problem, and some regularity results are concluded.

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“…The proof of Proposition 2.2 follows the usual techniques developed in [10] for the case of Sobolev spaces. For more details concerning the anisotropic Sobolev spaces, we refer the reader to [8] and [15].…”

Section: Preliminarymentioning

confidence: 99%

Existence of solutions for some quasilinear $\vec{p}(x)$-elliptic problem with Hardy potential

Azroul¹,

Bouziani²,

Hjiaj³

et al. 2018

Math.Boh.

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An Approximation Result in Generalized Anisotropic Sobolev Spaces and Applications

Abstract: In this paper, we give an approximation result in some anisotropic Sobolev space. We also describe the action of some distributions in the dual and we mention two applications to some strongly nonlinear anisotropic elliptic boundary value problems.

Cited by 27 publications

References 12 publications

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with -Laplacian

Existence of Renormalized Solutions for Some Anisotropic Quasilinear Elliptic Equations

Existence of solutions for some quasilinear $\vec{p}(x)$-elliptic problem with Hardy potential

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