Laplacian-like Operators Research Articles

A p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-Like Operators With Dirichlet Boundary Condition

This paper deals with a class of p(x)-Kirchhoff type problems involving the p(x)-Laplacian-like operators, arising from the capillarity phenomena, depending on two real parameters with Dirichlet boundary conditions. Using a topological degree for a class of demicontinuous operators of generalized (S+), we prove the existence of weak solutions of this problem. Our results extend and generalize several corresponding results from the existing literature. Keywords: p(x)-Kirchhoff type problems, p(x)-Laplacian-like operators, weak solutions, variable exponent Sobolev spaces.

Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove theexistence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

Abstract In this paper we study a Neumann boundary value problem of a new p(x)-Kirchhoff type problems driven by p(x)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (S+) type,weprove theexistenceofaweak solutionsof this problem. Our results are a natural generalisation of some existing ones in the context of p(x)-Kirchhoff type problems.

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Variable exponent $p(\cdot)$-Kirchhoff type problem with convection in variable exponent Sobolev spaces

We establish the existence of weak solution for a class of $p(x)$-Kirchhoff type problem for the $p(x)$-Laplacian-like operators with Dirichlet boundary condition and with gradient dependence (convection) in the reaction term. Our result is obtained using the topological degree for a class of demicontinuous operators of generalized $(S_{+})$ type and the theory of the variable exponent Sobolev spaces. Our results extend and generalize several corresponding results from the existing literature.

Weak solution of a Neumann boundary value problem with 𝑝(𝑥)-Laplacian-like operator

Abstract In this paper, we study the existence of a weak solution for a class of Neumann boundary value problems for equations involving the p ⁢ ( x ) p(x) -Laplacian-like operator. Using a topological degree theory for a class of demicontinuous operators of generalized ( S + ) (S_{+}) -type and the theory of the variable exponent Sobolev spaces, we establish the existence of a weak solution of this problem. Our results extend and generalize several corresponding results from the existing literature.

On a class of p(x)-Laplacian-like Dirichlet problem depending on three real parameters

This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: -Δp(x)lu+δ|u|α(x)-2u=μg(x,u)+λf(x,u,∇u)inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\displaystyle \\left\\{ \\begin{array}{ll} \\displaystyle -\\Delta ^{l}_{p(x)}u+\\delta \\vert u\\vert ^{\\alpha (x)-2}u=\\mu g(x, u)+\\lambda f(x, u, \ abla u) &{} \\mathrm {i}\\mathrm {n}\\ \\Omega ,\\\\ \\\\ u=0 &{} \\mathrm {o}\\mathrm {n}\\ \\partial \\Omega , \\end{array}\\right. \\end{aligned}$$\\end{document}where Delta ^{l}_{p(x)} is the p(x)-Laplacian-like operator, Omega is a smooth bounded domain in mathbb {R}^{N}, delta ,mu , and lambda are three real parameters, and p(cdot ),alpha (cdot )in C_{+}(overline{Omega }) and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized (S_{+}) type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.

Existence result for Neumann problems with p(x)-Laplacian-like operators in generalized Sobolev spaces

Existence result for Neumann problems with p(x)-Laplacian-like operators in generalized Sobolev spaces

P(x)-Laplacian-like Neumann problems in variable-exponent Sobolev spaces via topological degree methods

In this paper, we investigate the existence of a ?weak solutions? for a Neumann problems of p(x)-Laplacian-like operators, originated from a capillary phenomena, of the following form {?div( |?u|p(x)?2?u + |?u|2p(x)?2?u /?1 + |?u|2p(x))= ?f (x, u,?u) in ?,(|?u|p(x)?2?u + |?u|2p(x)?2 ?u/ ?|?u|2p(x)) ?u/?? = 0 on ??, in the setting of the variable-exponent Sobolev spaces W1,p(x)(?), where ? is a smooth bounded domain in RN, p(x) ? C+(??) and ? is a real parameter. Based on the topological degree for a class of demicontinuous operators of generalized (S+) type and the theory of variable-exponent Sobolev spaces, we obtain a result on the existence of weak solutions to the considered problem.

Ambrosetti\u2013Prodi Type Results for Dirichlet Problems of Fractional Laplacian-Like Operators

We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.

Infinitely Many Solutions for p(x)-Laplacian-Like Neumann Problems with Indefinite Weight

In the present paper, in view of the variational approach, we discuss the Neumann problems with indefinite weight and p(x)-Laplacian-like operators, originated from a capillary phenomena. Under certain assumptions, we prove the existence of infinitely many nontrival solutions of the problem.

Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator

Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator

A monolithic homotopy continuation algorithm with application to computational fluid dynamics

A new class of homotopy continuation methods is developed suitable for globalizing quasi-Newton methods for large sparse nonlinear systems of equations. The new continuation methods, described as monolithic homotopy continuation, differ from the classical predictor–corrector algorithm in that the predictor and corrector phases are replaced with a single phase which includes both a predictor and corrector component. Conditional convergence and stability are proved analytically. Using a Laplacian-like operator to construct the homotopy, the new algorithm is shown to be more efficient than the predictor–corrector homotopy continuation algorithm as well as an implementation of the widely-used pseudo-transient continuation algorithm for some inviscid and turbulent, subsonic and transonic external aerodynamic flows over the ONERA M6 wing and the NACA 0012 airfoil using a parallel implicit Newton–Krylov finite-difference flow solver.

On the superlinear problems involving [formula omitted]-Laplacian-like operators without AR-condition

In the present paper, in view of the variational approach, we discuss the nonlinear eigenvalue problems for p(x)-Laplacian-like operators, originated from a capillary phenomenon. Under some suitable conditions, we prove the existence of nontrivial solutions of the system for every parameter λ>0.

On the ground state of some fractional Laplacian-like operators

On the ground state of some fractional Laplacian-like operators

ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITION

Abstract. This paper deals with the superlinear elliptic problem with-out Ambrosetti and Rabinowitz type growth condition of the form:−div(1 + |∇u| p(x) √ 1+|∇u| 2p(x) )|∇u| p(x)−2 ∇u= λf(x,u), a.e. in Ω,u = 0, on ∂Ω,where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, λ > 0 isa parameter. The purpose of this paper is to obtain the existence resultsof nontrivial solutions for every parameter λ. Firstly, by using the moun-tain pass theorem a nontrivial solution is constructed for almost everyparameter λ > 0. Then we consider the continuation of the solutions.Our results are a generalization of that of Manuela Rodrigues. 1. IntroductionDuring the last fifteen years, variational problems and partial differentialequations with various types of nonstandard growth conditions have becomeincreasingly popular. This is partly due to their frequent appearance in ap-plications such as the modeling of electrorheological fluids [1, 12] and imageprocessing [2], but these problems are very interesting from a purely mathe-matical point of view as well.In this paper, we consider the following nonlinear eigenvalue problems forp(x)-Laplacian-like operators originated from a capillary phenomena of thefollowing form:(P)−div(1+

Multiple solutions to a class of inclusion problems with operator involvingp(x)-Laplacian

In this paper, we prove the existence of at least two nontrivial solutions for a non- linear elliptic problem involving p(x)-Laplacian-like operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions. p(x) p 1+jruj2p(x) )jruj p(x) 2 ru 2 @F (x;u); a:e: in ;

Multiplicity of Solutions on a Nonlinear Eigenvalue Problem for p(x)-Laplacian-like Operators

The paper study the existence and multiplicity of solutions for the nonlinear eigenvalue problems for p(x)-Laplacian-like operators, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with p-Laplacian-Like Operators

We investigate the following nonlinear equations with -Laplacian-like operators : some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin's continuation theorem. Our results are new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodic solutions for a kind of Lienard type -Laplacian equation, Nonlinear Analysis TMA, 69, 724–729, 2008).

Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators

Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with -Laplacian-Like Operators