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article

On optimal and quasi-optimal controls in coeﬃcients for multi-dimensional thermistor problem with mixed Dirichlet-Neumann boundary conditions

Peter I. Kogut

Control and Cybernetics > 2019 > Vol. 48, No. 1 > 31--68

In this paper we deal with an optimal control problem in coeﬃcients for the system of two coupled elliptic equations, also known as the thermistor problem, which provides a simultaneous description of the electric ﬁeld u = u(x) and temperature θ(x). The coeﬃcients of the operator div (B(x)∇θ(x)) are used as the controls in L∞(Ω). The optimal control problem is to minimize the discrepancy between a...

article

Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents

Abdelkrim Moussaoui, Jean Vélin

Journal of Elliptic and Parabolic Equations > 2018 > 4 > 2 > 417-440

This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder’s fixed point Theorem. A Moser iteration procedure is also obtained for singular quasilinear systems involving variable exponents establishing a priori estimates and boundedness...

article

Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems

Nguyen Thanh Chung

Mediterranean Journal of Mathematics > 2018 > 15 > 4 > 1-14

We consider the p(x)-Laplacian Robin eigenvalue problem $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta _{p(x)}u = \lambda V(x) |u|^{q(x)-2}u, \quad x\in \Omega ,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu }+\beta (x)|u|^{p(x)-2}u=0,\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$ -Δp(x)u=λV(x)|u|q(x)-2u,x∈Ω,|∇u|p(x)-2∂u∂ν+β(x)|u|p(x)-2u=0,x∈∂Ω, where $$\Omega $$ Ω is...

article

Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms

Jingjing Liu, Patrizia Pucci, Haitao Wu, Qihu Zhang

Journal of Mathematical Analysis and Applications > 2018 > 457 > 1 > 944-977

In this paper we investigate boundary blow-up solutions of the problem{−Δp(x)u+f(x,u)=±K(x)|∇u|m(x) in Ω,u(x)→+∞as d(x,∂Ω)→0, where Δp(x)u=div(|∇u|p(x)−2∇u) is called the p(x)-Laplacian. Our results extend the previous work [25] of Y. Liang, Q.H. Zhang and C.S. Zhao from the radial case to the non-radial setting, and [46] due to Q.H. Zhang and D. Motreanu from the assumption that K(x)|∇u(x)|m(x) is...

article

A Neumann problem for the p(x)-Laplacian with p=1 in a subdomain

Yiannis Karagiorgos, Nikos Yannakakis

Journal of Mathematical Analysis and Applications > 2017 > 454 > 1 > 412-428

In this paper we study a Neumann problem with non-homogeneous boundary conditions for the p(x)-Laplacian. In particular we assume that p(⋅) is a step function defined in a domain Ω and equals to 1 in a subdomain Ω1 and 2 in its complementary Ω2. By considering a suitable sequence pk of variable exponents such that pk→p and replacing p with pk in the original problem, we prove the existence of a solution...

article

Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

Annegret Glitzky, Matthias Liero

Nonlinear Analysis: Real World Applications > 2017 > 34 > C > 536-562

We study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current–voltage laws and self-heating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency...

article

Quasi-Newton minimization for the p(x)-Laplacian problem

M. Caliari, S. Zuccher

Journal of Computational and Applied Mathematics > 2017 > 309 > C > 122-131

We propose a quasi-Newton minimization approach for the solution of the p(x)-Laplacian elliptic problem, x∈Ω⊂Rm. This method outperforms those existing for the p(x)-variable case, which are based on general purpose minimizers such as BFGS. Moreover, when compared to ad hoc techniques available in literature for the p-constant case, and usually referred to as “mesh independent”, the present method...

article

An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

Damian Wiśniewski, Mariusz Bodzioch

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica > 2016 > 15

We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

article

The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior

Claudianor O. Alves, Sergey Shmarev, Jacson Simsen, Mariza S. Simsen

Journal of Mathematical Analysis and Applications > 2016 > 443 > 1 > 265-294

The paper addresses the questions of existence and asymptotic behavior of solutions to the Cauchy problem for the equationut−div(D(x)|∇u|p(x)−2∇u)+A(x)|u|q(x)−2u=f(x,t,u). The coefficients D, A are nonnegative functions which may vanish on a set of zero measure in Rn, and A(x)→∞ as |x|→∞, f(x,t,u) is globally Lipschitz with respect to u. The exponents p,q:Rn↦(1,∞) are given measurable functions. We...

article

On W1,q(⋅)-estimates for elliptic equations of p(x)-Laplacian type

Sun-Sig Byun, Jihoon Ok

Journal de Math�matiques Pures et Appliqu�es > 2016 > 106 > 3 > 512-545

We study nonlinear elliptic equations of p(x)-Laplacian type on nonsmooth domains to obtain an optimal Calderón–Zygmund type estimate in the variable exponent spaces. We find a correct regularity assumption on p(⋅), a minimal regularity requirement on the associated nonlinearity and a suitable flatness condition on the boundary of the underlying domain for such W1,q(⋅) regularity theory to hold true...

article

Existence and multiplicity of positive solutions to quasilinear elliptic equations with two parameters

Jeongmi Jeong, Chan-Gyun Kim

Afrika Matematika > 2017 > 28 > 1-2 > 237-247

In this paper, we are concerned with quasilinear elliptic equations with a Dirichlet boundary condition and two parameters. Using the truncation techniques and variational methods, the existence and/or multiplicity of positive solutions to the problems with a nonlinearity which satisfies either logistic growth rate or strong Allee effect growth rate are investigated. Also, a nonexistence result for...

article

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

Ky Ho, Inbo Sim

Science China Mathematics > 2017 > 60 > 1 > 133-146

We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones...

article

On some multiple solutions for a p(x)-Laplacian equation with critical growth

João Pablo P. da Silva

Journal of Mathematical Analysis and Applications > 2016 > 436 > 2 > 782-795

In this work we prove that, for each m∈N, if λ is small, there are m solutions for the equation −div(|∇u|p(x)−2∇u)=f(x,u)+λ|u|q(x)−2u, x∈Ω, in a bounded domain of RN, the nonlinearity is given by the critical growth in the context of variable exponents p⁎(x)=Np(x)/(N−p(x)). The main tools used are the variational method and concentration compactness principle.

article

On degenerate p(x)-Laplace equations involving critical growth with two parameters

Ky Ho, Inbo Sim

Nonlinear Analysis: Theory, Methods & Applications > 2016 > 132 > C > 95-114

After establishing a concentration-compactness principle for weighted variable exponent spaces, we show the existence of a nonnegative solution and infinitely many solutions for degenerate p(x)-Laplace equations involving critical growth with two parameters using various techniques in Calculus of Variations.

article

Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities

Giuseppina Barletta, Antonia Chinnì, Donal O’Regan

Nonlinear Analysis: Real World Applications > 2016 > 27 > C > 312-325

In this paper the existence of a nontrivial solution to a parametric Neumann problem for a class of nonlinear elliptic equations involving the p(x)-Laplacian and a discontinuous nonlinear term is established. Under a suitable condition on the behavior of the potential at 0+, we obtain an interval ]0,λ∗], such that, for any λ∈]0,λ∗] our problem admits at least one nontrivial weak solution. The solution...

article

Corrigendum to “Existence and some properties of solutions for degenerate elliptic equations with exponent variable” [Nonlinear Anal. 98 (2014) 146–164]

Ky Ho, Inbo Sim

Nonlinear Analysis: Theory, Methods & Applications > 2015 > 128 > C > 423-426

We restate Theorem 4.2 and Lemma 4.3 in [1] a little differently and give a proof of Lemma 4.3 and replace corresponding parts in the proof of Theorem 4.2.

article

Existence results for a class of p(x)-Laplacian problems in RN

Mostafa Allaoui, Abdelrachid El Amrouss, Anass Ourraoui

Computers and Mathematics with Applications > 2015 > 69 > 7 > 582-591

This paper shows the existence of solutions for problem involving the p(x)-Laplacian in RN with an indefinite weights a and b, by means of the Sobolev embedding theorem and under some various conditions on the nonlinearity.

article

Continuous spectrum of Steklov nonhomogeneous elliptic problem

M. Allaoui

Opuscula Mathematica > 2015 > Vol. 35, no. 6 > 853--866

By applying two versions of the mountain pass theorem and Ekeland's variational principle, we prove three different situations of the existence of solutions for the following Steklov problem: [formula] where Ω ⊂ RN (N ≥ 2) is a bounded smooth domain and p, q: Ω → (1, + ∞) are continuous functions.

article

Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent

Gabriele Bonanno, Antonia Chinnì

Journal of Mathematical Analysis and Applications > 2014 > 418 > 2 > 812-827

Under appropriate growth conditions on the nonlinearity, the existence of multiple solutions for nonlinear elliptic Dirichlet problems with variable exponent is established. The approach is based on variational methods. Some applications and examples illustrate the obtained results.

article

Existence of entire solutions for a class of variable exponent elliptic equations

Patrizia Pucci, Qihu Zhang

Journal of Differential Equations > 2014 > 257 > 5 > 1529-1566

The paper deals with the existence of entire solutions for a quasilinear equation (Eλ) in RN, depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value λ⁎>0 with the property that (Eλ) admits nontrivial...

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On optimal and quasi-optimal controls in coeﬃcients for multi-dimensional thermistor problem with mixed Dirichlet-Neumann boundary conditions

Existence and a priori estimates of solutions for quasilinear singular elliptic systems with variable exponents

Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems

Existence and blow-up rate of large solutions of p(x)-Laplacian equations with gradient terms

A Neumann problem for the p(x)-Laplacian with p=1 in a subdomain

Analysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices

Quasi-Newton minimization for the p(x)-Laplacian problem

An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior

On W1,q(⋅)-estimates for elliptic equations of p(x)-Laplacian type

Existence and multiplicity of positive solutions to quasilinear elliptic equations with two parameters

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

On some multiple solutions for a p(x)-Laplacian equation with critical growth

On degenerate p(x)-Laplace equations involving critical growth with two parameters

Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities

Corrigendum to “Existence and some properties of solutions for degenerate elliptic equations with exponent variable” [Nonlinear Anal. 98 (2014) 146–164]

Existence results for a class of p(x)-Laplacian problems in RN

Continuous spectrum of Steklov nonhomogeneous elliptic problem

Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent

Existence of entire solutions for a class of variable exponent elliptic equations

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