scholarly journals Quasilinear degenerate elliptic unilateral problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 11-31 ◽  
Author(s):  
L. Aharouch ◽  
Y. Akdim ◽  
E. Azroul
Keyword(s):  
Lower Order ◽  
Existence Result ◽  
Order Term ◽  
Natural Growth ◽  
Unilateral Problem ◽  
Sign Condition ◽  
Carathéodory Function ◽  
Degenerate Elliptic ◽  
Unilateral Problems ◽  
The Right

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the formAu+H(x,u,∇u)=f, whereAis a Leray-Lions operator fromW1,p(Ω,w)into its dual. On the nonlinear lower-order termH(x,u,∇u), we assume that it is a Carathéodory function having natural growth with respect to|∇u|, but without assuming the sign condition. The right-hand sidefbelongs toL1(Ω).

scholarly journals Strongly nonlinear elliptic variational unilateral problems in Orlicz space

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
L. Aharouch ◽  
A. Benkirane ◽  
M. Rhoudaf
Keyword(s):  
Existence Result ◽  
Order Term ◽  
Natural Growth ◽  
Unilateral Problem ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Sign Condition ◽  
Carathéodory Function ◽  
Unilateral Problems ◽  
The Right

We will be concerned with the existence result of unilateral problem associated to the equations of the formAu+g(x,u,∇u)=f, whereAis a Leray-Lions operator from its domainD(A)⊂W01LM(Ω)intoW−1EM¯(Ω). On the nonlinear lower order termg(x,u,∇u), we assume that it is a Carathéodory function having natural growth with respect to|∇u|, and satisfies the sign condition. The right-hand sidefbelongs toW−1EM¯(Ω).

Existence results for diffusion–convection equations of nonlinear unilateral problems defined in Orlicz–Sobolev spaces

2018 ◽  
Vol 11 (06) ◽  
pp. 1850079
Author(s):  
H. Moussa ◽  
M. Rhoudaf ◽  
H. Sabiki
Keyword(s):  
Elliptic Equations ◽  
Order Term ◽  
Nonlinear Elliptic Equations ◽  
Strongly Nonlinear ◽  
Nonlinear Elliptic ◽  
Polynomial Type ◽  
Carathéodory Function ◽  
Unilateral Problems ◽  
The Right ◽  
Diffusion Convection

We prove the existence result of unilateral problems associated to strongly nonlinear elliptic equations whose model, including the diffusion–convection equation, is [Formula: see text]. We study exactly the following general case [Formula: see text] where [Formula: see text] is a Leray–Lions operator having a growth not necessarily of polynomial type, the lower order term [Formula: see text] : [Formula: see text] is a Carathéodory function, for a.e. [Formula: see text] and for all [Formula: see text] satisfying only a growth condition and the right-hand side [Formula: see text] belongs to [Formula: see text].

scholarly journals Existence results for a perturbed Dirichlet problem without sign condition in Orlicz spaces

2020 ◽  
Vol 72 (4) ◽  
pp. 509-526
Author(s):  
H. Moussa ◽  
M. Rhoudaf ◽  
H. Sabiki
Keyword(s):  
Elliptic Equations ◽  
Existence Result ◽  
Orlicz Spaces ◽  
Nonlinear Elliptic Equations ◽  
Existence Results ◽  
Nonlinear Elliptic ◽  
Sign Condition ◽  
Carathéodory Function ◽  
Monotone Vector Field ◽  
The Right

UDC 517.5 We deal with the existence result for nonlinear elliptic equations related to the form < b r > A u + g ( x , u , ∇ u ) = f , < b r > where the term - ⅆ i v ( a ( x , u , ∇ u ) ) is a Leray–Lions operator from a subset of W 0 1 L M ( Ω ) into its dual.  The growth and coercivity conditions on the monotone vector field a are prescribed by an N -function M which does not have to satisfy a Δ 2 -condition. Therefore we use Orlicz–Sobolev spaces which are not necessarily reflexive and assume that the nonlinearity g ( x , u , ∇ u ) is a Carathéodory function satisfying only a growth condition with no sign condition. The right-hand side~ f belongs to W -1 E M ¯ ( Ω ) .

scholarly journals Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

2018 ◽  
Vol 4 (2) ◽  
pp. 171-188 ◽  
Author(s):  
Youssef Akdim ◽  
Chakir Allalou ◽  
Abdelhafid Salmani
Keyword(s):  
Lower Order ◽  
Existence Of Solutions ◽  
Entropy Solutions ◽  
Unilateral Problem ◽  
Nonlinear Elliptic ◽  
Right Hand ◽  
Unilateral Problems ◽  
The Right ◽  
Lower Order Terms

AbstractIn this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$where the right hand side f belongs to L1(Ω). The operator $- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $ is a Leray-Lions anisotropic operator and ϕi ∈ C0(ℝ,ℝ).

scholarly journals Nonlinear evolution problems with singular coefficients in the lower order terms

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Lower Order ◽  
Time Horizon ◽  
Existence Result ◽  
Nonlinear Evolution ◽  
Order Term ◽  
Time Behavior ◽  
Infinite Time ◽  
Singular Coefficient ◽  
Singular Coefficients ◽  
Lower Order Terms

AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.

scholarly journals Anisotropic nonlinear problem of infinite order with variables exponents and $L^1$ data

2020 ◽  
Author(s):  
Moussa Chrif ◽  
hakima ouyahya
Keyword(s):  
Nonlinear Equation ◽  
Elliptic Operator ◽  
Nonlinear Problem ◽  
Sobolev Spaces ◽  
Lower Order ◽  
Infinite Order ◽  
Existence Of Solutions ◽  
Order Term ◽  
Strongly Nonlinear ◽  
Sign Condition

In this paper, we prove the existence of solutions for the strongly nonlinear equation of the type $$Au+g(x,u)=f$$ where $A$ is an elliptic operator of infinite order from a functional Sobolev spaces of infinite order with variables exponents to its dual. $g(x, s)$ is a lower order term satisfying essentially a sign condition on s and the second term f belongs to $L^1(\Omega)$.

scholarly journals Non-linear elliptic unilateral problems in Musielak-Orlicz spaces with L1 data

2018 ◽  
Vol 36 (1) ◽  
pp. 51
Author(s):  
Mustafa Ait Khellou ◽  
Abdelmoujib Benkirane
Keyword(s):  
Sobolev Space ◽  
Existence Result ◽  
Orlicz Spaces ◽  
Conjugate Function ◽  
Natural Growth ◽  
Nonlinear Elliptic ◽  
L1 Data ◽  
Non Linear ◽  
Unilateral Problems ◽  
Natural Growth Terms

We prove an existence result of solutions for nonlinear elliptic unilateral problems having natural growth terms and L1 data in Musielak-Orlicz-Sobolev space W1Lφ, under the assumption that the conjugate function of φ satisfies the ∆2-condition.

Relaxation of Ginzburg–Landau functional perturbed by continuous nonlinear lower-order term in one dimension

2014 ◽  
Vol 13 (01) ◽  
pp. 101-123 ◽  
Author(s):  
Andrija Raguž
Keyword(s):  
Asymptotic Behavior ◽  
Lower Order ◽  
Order Term ◽  
Lower Order Term ◽  
One Dimension ◽  
Young Measures ◽  
Ginzburg Landau ◽  
Carathéodory Function ◽  
Γ Convergence

We study the asymptotic behavior as ε → 0 of the Ginzburg–Landau functional [Formula: see text], where A(s, v, v′) is the nonlinear lower-order term generated by certain Carathéodory function a : (0, 1)2 × R2 → R. We obtain Γ-convergence for the rescaled functionals [Formula: see text] as ε → 0 by using the notion of Young measures on micropatterns, which was introduced in 2001 by Alberti and Müller. We prove that for ε ≈ 0 the minimal value of [Formula: see text] is close to [Formula: see text], where A∞(s) : = ½A(s, 0, -1) + ½A(s, 0, 1) and where E0 depends only on W. Further, we use this example to establish some general conclusions related to the approach of Alberti and Müller.

scholarly journals Existence of entropy solutions for degenerate elliptic unilateral problems with variable exponents

2018 ◽  
Vol 36 (1) ◽  
pp. 79
Author(s):  
Elhoussine Azroul ◽  
Abdelkrim Barbara ◽  
Mohamed Badr Benboubker ◽  
Khalid El Haiti
Keyword(s):  
Variable Exponent ◽  
Nonlinear Term ◽  
Weighted Sobolev Spaces ◽  
Positive Function ◽  
Variable Exponents ◽  
Integrability Conditions ◽  
Degenerate Elliptic ◽  
Unilateral Problems ◽  
The Right ◽  
Strictly Positive Function

In this article, we study the following degenerate unilateral problems:  $$ -\mbox{ div} (a(x,\nabla u))+H(x,u,\nabla u)=f,$$ which is subject to the Weighted Sobolev spaces with variable exponent $W^{1,p(x)}_{0}(\Omega,\omega)$, where $\omega$ is a weight function on $\Omega$, ($\omega$ is a measurable, a.e. strictly positive function on $\Omega$ and satisfying some integrability conditions). The function $H(x,s,\xi)$ is a nonlinear term satisfying some growth condition but no sign condition  and the right hand side $f\in L^1(\Omega)$.

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