Nonlocal approximations to anisotropic Sobolev norms

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent p ( x ) reaches the value 1.

Existence Results for a Class of p(x)-Kirchhoff Problems

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:Where and satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

Nonlocal characterizations of variable exponent Sobolev spaces

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.

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

The 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.

Existence of renormalized solutions for some quasilinear elliptic Neumann problems

This paper is devoted to study some nonlinear elliptic Neumann equations of the type { A u + g ( x , u , ∇ u ) + | u | q ( ⋅ ) - 2 u = f ( x , u , ∇ u ) in Ω , ∑ i = 1 N a i ( x , u , ∇ u ) ⋅ n i = 0 on ∂ Ω , \left\{ {\matrix{ {Au + g(x,u,\nabla u) + |u{|^{q( \cdot ) - 2}}u = f(x,u,\nabla u)} \hfill & {{\rm{in}}} \hfill & {\Omega ,} \hfill \cr {\sum\limits_{i = 1}^N {{a_i}(x,u,\nabla u) \cdot {n_i} = 0} } \hfill & {{\rm{on}}} \hfill & {\partial \Omega ,} \hfill \cr } } \right. in the anisotropic variable exponent Sobolev spaces, where A is a Leray-Lions operator and g(x, u, ∇u), f (x, u, ∇u) are two Carathéodory functions that verify some growth conditions. We prove the existence of renormalized solutions for our strongly nonlinear elliptic Neumann problem.

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

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.

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

The 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).

Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L1-data

In the present paper, we prove existence results of entropy solutions to a class of nonlinear degenerate parabolic p(·)-Laplacian problem with Dirichlet-type boundary conditions and L1 data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.