Mostafa El Moumni

Existence result for strongly nonlinear elliptic unilateral problems with L 1-data

We prove the existence of a solution of unilateral problems associated to strongly nonlinear elliptic equations whose model, including the stationary convection-diffusion equation, is with ρ ≥ 0, σ > 0, the function |b| has a suitable summability and the right-hand side f belongs to L 1(Ω).

Existence of renormalized solutions for some strongly nonlinear elliptic equations in Orlicz spaces

Abstract We prove the existence of a renormalized solution for the Dirichlet problem associated to the nonlinear elliptic equations div a(x,u,∇u) - div Φ(u) + g(x,u,∇u) = f in Ω, in the general setting of Orlicz spaces, where Φ ∈ 𝒞0(ℝ,ℝN), the function g has a natural growth with respect to its third argument and satisfies the sign condition while the datum f belongs to W -1EM¯(Ω). No Δ2-condition is needed for the considered N-functions.

An Approximation of Hedberg’s Type in Sobolev Spaces with Variable Exponent and Application

The aim of this paper is to extend the usual framework of PDE with to include a large class of cases with , whose coefficient satisfies conditions (including growth conditions) which guarantee the solvability of the problem . This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.

p ( x ) p(x) -biharmonic operator involving the p ( x ) p(x) -Hardy inequality

Abstract In this work, we investigate the spectrum denoted by Λ for the p ⁢ ( x ) {p(x)} -biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup ⁡ Λ = + ∞ {\sup\Lambda=+\infty} . Moreover, we prove that inf ⁡ Λ > 0 {\inf\Lambda>0} if and only if the domain Ω satisfies the p ⁢ ( x ) {p(x)} -Hardy inequality.

Existence of renormalized solutions for strongly nonlinear parabolic problems with measure data

Abstract We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ ⁡ b ⁢ ( x , u ) ∂ ⁡ t - div ⁡ ( a ⁢ ( x , t , u , ∇ ⁡ u ) ) + g ⁢ ( x , t , u , ∇ ⁡ u ) + H ⁢ ( x , t , ∇ ⁡ u ) = μ   in ⁢ Ω × ( 0 , T ) ,

BOUNDED SOLUTIONS OF UNILATERAL PROBLEMS FOR STRONGLY NONLINEAR EQUATIONS IN ORLICZ SPACES

{\partial b(x,u)\over\partial t}-\operatorname{div}(a(x,t,u,\nabla u))+g(x,t,u% ,\nabla u)+H(x,t,\nabla u)=\mu\quad\text{in }\Omega\times(0,T),

Entropy solution for strongly nonlinear elliptic problems with lower order terms and

where the right-hand side belongs to L 1 ⁢ ( Q T ) + L p ′ ⁢ ( 0 , T ; W - 1 , p ′ ⁢ ( Ω ) )

L^1

{L^{1}(Q_{T})+L^{p^{\prime}}(0,T;W^{-1,p^{\prime}}(\Omega))}

-data

and b ⁢ ( x , u )

Solutions of nonlinear elliptic problems with lower order terms

{b(x,u)}