scholarly journals Infinitely Many Solutions of the Neumann Problem for Elliptic systems in Anisotropic Variable Exponent Sobolev Spaces

2017 ◽  
Vol 3 (1) ◽  
pp. 70-82
Author(s):  
A. Ahmed ◽  
M.S.B. Elemine Vall ◽  
A. Touzani
Keyword(s):  
Variational Principle ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Elliptic Systems ◽  
Subject Classification ◽  
Infinitely Many Solutions ◽  
Variable Exponent Sobolev Spaces ◽  
The Neumann Problem

Abstract In this paper, we prove the existence of in finitely many solutions for the following system by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces 2010 Mathematics Subject Classification. 35K05 - 35K55.

scholarly journals Infinitely many solutions for a nonlinear Navier boundary systems involving $(p(x),q(x))$-biharmonic

2014 ◽  
Vol 33 (1) ◽  
pp. 155
Author(s):  
Mostafa Allaoui ◽  
Abdel Rachid El Amrouss ◽  
Anass Ourraoui
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Type System ◽  
Infinitely Many Solutions ◽  
Variable Exponent Sobolev Spaces

In this article, we study the following $(p(x),q(x))$-biharmonic type system \begin{gather*} \Delta(|\Delta u|^{p(x)-2}\Delta u)=\lambda F_u(x,u,v)\quad\text{in }\Omega,\\ \Delta(|\Delta v|^{q(x)-2}\Delta v)=\lambda F_v(x,u,v)\quad\text{in }\Omega,\\ u=v=\Delta u=\Delta v=0\quad  \text{on }\partial\Omega. \end{gather*} We prove the existence of infinitely many solutions of the problem byapplying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

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 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 Multiple Solutions for Nonlinear Navier Boundary Systems Involving(p1(x),…,pn(x))-Biharmonic Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Multiple Solutions ◽  
Variable Exponent ◽  
Multiplicity Of Solutions ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Variable Exponent Sobolev Spaces

We improve some results on the existence and multiplicity of solutions for the(p1(x),…,pn(x))-biharmonic system. Our main results are new. Our approach is based on general variational principle and the theory of the variable exponent Sobolev spaces.

scholarly journals Infinitely many solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions

2019 ◽  
Vol 38 (4) ◽  
pp. 71-96 ◽  
Author(s):  
Shapour Heidarkhani ◽  
Anderson Luis Albuquerque de Araujo ◽  
Amjad Salari
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variable Exponent ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Nonlocal Problems ◽  
Multiplicity Results

In this article we will provide new multiplicity results of the solutions for nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. We investigate the existence of infinitely many solutions for perturbed nonlocal problems with variable exponent and nonhomogeneous Neumann conditions. The approach is based on variational methods and critical point theory.

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