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].

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 The Existence of Solutions to the NonhomogeneousA-Harmonic Equations with Variable Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Haiyu Wen
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Obstacle Problem ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Variable Exponent Sobolev Space ◽  
Harmonic Equation ◽  
Weighted Variable

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneousA-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equationd⋆A(x,dω)=B(x,dω)in the weighted variable exponent Sobolev spaceWdp(x)(Ω,Λl,μ).

scholarly journals On the boundary values of the solutions of linear elliptic equations

1983 ◽  
Vol 27 (1) ◽  
pp. 1-30 ◽  
Author(s):  
J. Chabrowski ◽  
H.B. Thompson
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Condition ◽  
Boundary Values ◽  
Linear Elliptic Equation

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.

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 On compact and bounded embedding in variable exponent Sobolev spaces and its applications

2019 ◽  
Vol 9 (2) ◽  
pp. 401-414
Author(s):  
Farman Mamedov ◽  
Sayali Mammadli ◽  
Yashar Shukurov
Keyword(s):  
Sobolev Space ◽  
Growth Condition ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Lebesgue Spaces ◽  
Nonstandard Growth ◽  
Variable Exponent Sobolev Spaces ◽  
Nonstandard Growth Condition ◽  
Nonlinear Ode ◽  
Weighted Variable

Abstract For a weighted variable exponent Sobolev space, the compact and bounded embedding results are proved. For that, new boundedness and compact action properties are established for Hardy’s operator and its conjugate in weighted variable exponent Lebesgue spaces. Furthermore, the obtained results are applied to the existence of positive eigenfunctions for a concrete class of nonlinear ode with nonstandard growth condition.

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