scholarly journals Existence and uniqueness of weak solution of p(x)- Laplacian in Sobolev spaces with variable exponents in complete manifolds

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1453-1463
Author(s):  
Omar Benslimane ◽  
Ahmed Aberqi ◽  
Jaouad Bennouna
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Polynomial Growth ◽  
Mountain Pass Theorem ◽  
Variable Exponents ◽  
Variable Exponent Sobolev Spaces ◽  
Complete Manifolds ◽  
Laplacian Equations

The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous p(x)- Laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.

Sobolev spaces with variable exponents on Riemannian manifolds

2013 ◽  
Vol 92 ◽  
pp. 47-59 ◽  
Author(s):  
Michał Gaczkowski ◽  
Przemysław Górka
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Variable Exponents

scholarly journals Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

2021 ◽  
Vol 7 (1) ◽  
pp. 50-65
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Problem ◽  
Topological Degree ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Technical Approach ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
Topological Degree Method

AbstractThe aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W−1,p′(.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

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