Low-dimensional compact embeddings of symmetric Sobolev spaces with applications

2011 ◽  
Vol 141 (2) ◽  
pp. 383-395 ◽  
Author(s):  
Francesca Faraci ◽  
Antonio Iannizzotto ◽  
Alexandru Kristály
Keyword(s):  
Sobolev Space ◽  
Neumann Problem ◽  
Unbounded Domain ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Symmetric Functions ◽  
Laplacian Operator ◽  
Compact Embeddings ◽  
Cylindrically Symmetric ◽  
Low Dimensional

If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L∈(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∈(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

scholarly journals Three weak solutions for a Neumann elliptic equations involving the p→(x)\vec p\left( x \right)-Laplacian operator

2020 ◽  
Vol 7 (1) ◽  
pp. 224-236
Author(s):  
Ahmed Ahmed ◽  
Mohamed Saad Bouh Elemine Vall
Keyword(s):  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Laplacian Operator ◽  
Variable Exponent Sobolev Spaces

AbstractThe aim of this paper is to establish the existence of at least three weak solutions for the following elliptic Neumann problem \left\{ {\matrix{ { - {\Delta _{\vec p\left( x \right)}}u + \alpha \left( x \right){{\left| u \right|}^{{p_0}\left( x \right) - 2}}u = \lambda f\left( {x,u} \right)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{{\left| {{{\partial u} \over {\partial {x_i}}}} \right|}^{{p_i}\left( x \right) - 2}}{{\partial u} \over {\partial {x_i}}}{\gamma _i} = 0} } \hfill & {on} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces W1,\vec p\left( \cdot \right)\left( \Omega \right) where λ > 0 and f (x, t) = |t|q(x)−2t − |t|s(x)−2t, x ∈ Ω, t ∈ 𝕉 and q(·), s\left( \cdot \right) \in {\mathcal{C}_ + }\left( {\bar \Omega } \right).

scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

Existence results for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Shapour Heidarkhani ◽  
Giuseppe Caristi ◽  
Ghasem A. Afrouzi ◽  
Shahin Moradi
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Sobolev Space ◽  
Banach Spaces ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Existence Results ◽  
Reflexive Banach Spaces

Abstract Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

2018 ◽  
Vol 68 (4) ◽  
pp. 867-880
Author(s):  
Saeid Shokooh ◽  
Ghasem A. Afrouzi ◽  
John R. Graef
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Neumann Problems

Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

scholarly journals Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Boundary Problem ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Ekeland’S Variational Principle ◽  
Laplacian Operator ◽  
Ekeland's Variational Principle ◽  
Existence Of Weak Solutions

We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator:  -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in  Ω,∂u/∂ν=0,on  ∂Ω. Under some appropriate conditions on the functionsp,  e,  a, and  f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.

scholarly journals Multiplicity of Solutions for Perturbed Nonhomogeneous Neumann Problem through Orlicz-Sobolev Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Liu Yang
Keyword(s):  
Sobolev Space ◽  
Variational Methods ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Multiple Solutions ◽  
Sufficient Conditions ◽  
Three Solutions ◽  
Multiplicity Of Solutions ◽  
Critical Point Theorems

We investigate the existence of multiple solutions for a class of nonhomogeneous Neumann problem with a perturbed term. By using variational methods and three critical point theorems of B. Ricceri, we establish some new sufficient conditions under which such a problem possesses three solutions in an appropriate Orlicz-Sobolev space.

scholarly journals Three solutions to a p(x)-Laplacian problem in weighted-variable-exponent Sobolev space

2013 ◽  
Vol 21 (2) ◽  
pp. 195-205 ◽  
Author(s):  
Wen-Wu Pan ◽  
Ghasem Alizadeh Afrouzi ◽  
Lin Li
Keyword(s):  
Sobolev Space ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Three Solutions ◽  
Variable Exponent Sobolev Space ◽  
Weighted Variable

Abstract In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567].

Compact embeddings of some weighted Sobolev spaces on ℝN

1995 ◽  
Vol 117 (2) ◽  
pp. 333-338 ◽  
Author(s):  
Raffaele Chiappinelli
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Compact Embeddings ◽  
Measurable Functions ◽  
Image Position

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

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