On a fourth order Dirichlet Problem

2010 ◽  
Vol 17 (3) ◽  
pp. 495-509
Author(s):  
Marek Galewski ◽  
Joanna Smejda
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Concave Function ◽  
Growth Conditions ◽  
Dirichlet Problems ◽  
Convex Part ◽  
Dual Variational Method ◽  
Derivatives Of

Abstract We consider by a dual variational method the existence of solutions to certain fourth order Dirichlet problems with nonlinearities corresponding to the derivatives of a sum of a convex and a concave function. The growth conditions are imposed only on the convex part.

A note on a dirichlet problem with concave-convex nonlinearity

2010 ◽  
Vol 60 (3) ◽  
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Convex Functions ◽  
Elliptic Problems ◽  
Growth Conditions ◽  
Abstract Result ◽  
Dual Variational Method ◽  
Existence Principle ◽  
Derivatives Of

AbstractWe prove an existence principle that would apply for elliptic problems with nonlinearity separating into a difference of derivatives of two convex functions in the case when the growth conditions are imposed only on the minuend term. We present abstract result and its application. We modify the so called dual variational method.

scholarly journals On the Nonlinear Dirichlet Problem with P(x)-Laplacian

2007 ◽  
Vol 75 (3) ◽  
pp. 381-395 ◽  
Author(s):  
Marek Galewski ◽  
Marek Płócienniczak
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Bounded Domain ◽  
Growth Conditions ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Dual Variational Method ◽  
Existence And Stability ◽  
Laplacian Operators

Using a dual variational method which we develop, we show the existence and stability of solutions for a family of Dirichlet problems k = 0, 1,… in a bounded domain in ℝN and with the nonlinearity satisfying some general growth conditions. The assumptions put on v are satisfied by p(x)-Laplacian operators.

Dual Variational Method for a Fourth Order Dirichlet Problem

2008 ◽  
Vol 15 (4) ◽  
pp. 653-664
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Fourth Order ◽  
Functional Parameter ◽  
Dirichlet Problems ◽  
Dual Variational Method ◽  
Existence And Stability ◽  
Dual Action ◽  
Primal And Dual ◽  
Stability Results

Abstract We obtain the existence and stability results for a fourth order Dirichlet problems with nonlinearity being convex in a certain interval. A dual variational method is introduced, which relies on investigating the primal and dual action functionals on certain subsets of their domains. The dependence on a functional parameter for a fourth order Dirichlet problem is considered as a consequence of stability.

On the Dirichlet problem with nonconvex nonlinearity

2010 ◽  
Vol 47 (2) ◽  
pp. 190-199
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Growth Conditions ◽  
Existence Results ◽  
Dirichlet Problems ◽  
Local Growth ◽  
Dual Variational Method ◽  
Nonlinear Eigenvalue

We provide existence results for 2 m order Dirichlet problems with nonconvex nonlinearity which satisfies general local growth conditions. In doing so we construct a dual variational method. Problem considered relates to the problem of nonlinear eigenvalue.

scholarly journals A new variational method for thep(x)-Laplacian equation

2005 ◽  
Vol 72 (1) ◽  
pp. 53-65 ◽  
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Existence Of Solutions ◽  
Laplacian Equation ◽  
Palais Smale Condition ◽  
Dual Variational Method

Using a dual variational method we shall show the existence of solutions to the Dirichlet problemwithout assuming Palais-Smale condition.

scholarly journals Variational Method to p-Laplacian Fractional Dirichlet Problem with Instantaneous and Noninstantaneous Impulses

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yiru Chen ◽  
Haibo Gu ◽  
Lina Ma
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Major Result ◽  
Dirichlet Boundary Value Problem ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p-Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result.

scholarly journals On the Dirichlet problem of elliptic type

2007 ◽  
Vol 75 (2) ◽  
pp. 169-177
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Existence Of Solutions ◽  
Elliptic Type ◽  
Numerical Parameter ◽  
Dirichlet Problems ◽  
Dual Action

We investigate the existence of solutions and their stability for elliptic Dirichlet problems with nonlinearity of a convex-concave type. By relating the primal action and the dual action functionals on certain subsets of their domains we get the existence of solutions which are further stable with respect to a numerical parameter. We allow also for the differential operator to depend on a numerical parameter.

scholarly journals Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

2016 ◽  
Vol 34 (1) ◽  
pp. 253-272
Author(s):  
Khalil Ben Haddouch ◽  
Zakaria El Allali ◽  
Najib Tsouli ◽  
Siham El Habib ◽  
Fouad Kissi
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Neumann Boundary ◽  
Growth Conditions ◽  
Neumann Boundary Conditions ◽  
Order Elliptic Equation ◽  
Fourth Order Eigenvalue Problem ◽  
Fourth Order Elliptic Equation

In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x)$-growth conditions $\Delta^2_{p(x)} u=\lambda |u|^{p(x)-2} u$, under Neumann boundary conditions, where $p(x)$ is a continuous function defined on the bounded domain with $p(x)>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\sup \Lambda =+\infty$, where $\Lambda$ is the set of all eigenvalues.

scholarly journals Existence of bounded solutions for fourth order Dirichlet problems with convex-concave nonlinearity

2009 ◽  
Vol 42 (1) ◽  
Author(s):  
Marek Galewski
Keyword(s):  
Boundary Value Problem ◽  
Growth Condition ◽  
Fourth Order ◽  
Dirichlet Boundary ◽  
Concave Function ◽  
Higher Order ◽  
Dirichlet Boundary Value Problem ◽  
Bounded Solutions ◽  
Dirichlet Problems ◽  
Concave Part

AbstractWe consider the Dirichlet boundary value problem for higher order O.D.E. with nonlinearity being the sum of a derivative of a convex and of a concave function in case when no growth condition is imposed on the concave part.

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