scholarly journals Variational Inequalities with Multivalued Lower Order Terms and Convex Functionals in Orlicz-Sobolev Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Ge Dong ◽  
Xiaochun Fang
Keyword(s):  
Sobolev Spaces ◽  
Lower Order ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Order Term ◽  
Convex Functional ◽  
Polynomial Type ◽  
Convex Functionals ◽  
Order Elliptic Operator ◽  
Lower Order Terms

We consider the existence of solutions of variational inequality form. Findu∈D(J):〈A(u),v-u〉+〈F(u),v-u〉+J(v)-J(u)≥0,∀v∈W1LM(Ω),whose principal part is having a growth not necessarily of polynomial type, whereAis a second-order elliptic operator of Leray-Lions type,Fis a multivalued lower order term, andJis a convex functional. We use subsupersolution methods to study the existence and enclosure of solutions in Orlicz-Sobolev spaces.

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

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 $L^1(\Omega)$.

On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

2018 ◽  
Vol 18 (2) ◽  
pp. 269-287 ◽  
Author(s):  
Vy Khoi Le ◽  
Klaus Schmitt
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Lower Order ◽  
Order Term ◽  
Growth Conditions ◽  
Principal Term ◽  
Lower Order Term ◽  
Convex Functional ◽  
Parabolic Variational Inequality ◽  
Convex Functionals

Abstract In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and {f\in F(\cdot,\cdot,u)} such that {u(\cdot,0)=u_{0}} and \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.

scholarly journals Nonlinear elliptic equations with variable exponents involving singular nonlinearity

2021 ◽  
Vol 8 (4) ◽  
pp. 705-715
Author(s):  
H. Khelifi ◽  
◽  
Y. El Hadfi ◽  
◽  
Keyword(s):  
Sobolev Spaces ◽  
Lower Order ◽  
Elliptic Equations ◽  
Order Term ◽  
Nonlinear Elliptic Equations ◽  
Variable Exponents ◽  
Singular Nonlinearity ◽  
Nonlinear Elliptic ◽  
Regularizing Effects ◽  
Lower Order Terms

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and L1 datum in the setting of Sobolev spaces with variable exponents. We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in [1–3].

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 Existence of solutions for some strongly nonlinear parabolic problems involving lower order terms in divergence form in Musielak-Orlicz spaces

2019 ◽  
Vol 38 (6) ◽  
pp. 99-126
Author(s):  
Abdeslam Talha ◽  
Abdelmoujib Benkirane
Keyword(s):  
Lower Order ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Orlicz Spaces ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Strongly Nonlinear ◽  
L1 Data ◽  
Nonlinear Parabolic ◽  
Lower Order Terms

In this work, we prove an existence result of entropy solutions in Musielak-Orlicz-Sobolev spaces for a class of nonlinear parabolic equations with two lower order terms and L1-data.

Variational Inequalities with General Multivalued Lower Order Terms Given by Integrals

2011 ◽  
Vol 11 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Vy Khoi
Keyword(s):  
Variational Inequalities ◽  
Lower Order ◽  
Order Term ◽  
Multivalued Function ◽  
Solution Set ◽  
Lower Order Term ◽  
Extremal Solutions ◽  
Lower Order Terms

AbstractThis paper is about the existence and some properties of solutions of variational inequalities associated with the 2nd order inclusiondiv[A(x, ∇u)] + L ∈ f (x, u) in Ω,where the lower order term f (x, u) is a general multivalued function. Both coercive and noncoercive cases are considered. In the noncoercive case, we use a sub-supersolution approach to study the existence, comparison, and other properties of the solution set such as its compactness, directedness, and the existence of extremal solutions.

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