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 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.

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.

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.

scholarly journals A note on the stability and the approximation of solutions for a Dirichlet problem with p(x)-Laplacian

2007 ◽  
Vol 49 (1) ◽  
pp. 75-83
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Growth Conditions ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Secondary 35B35 ◽  
Local Growth ◽  
Keywords And Phrases ◽  
2000 Mathematics Subject Classification ◽  
The Stability ◽  
Stability Results

We show the stability results and Galerkin-type approximations of solutions for a family of Dirichlet problems with nonlinearity satisfying some local growth conditions. 2000 Mathematics subject classification: primary 35A15; secondary 35B35, 65N30. Keywords and phrases: p(x)-Laplacian, duality, variational method, stability of solutions, Galerkin-type approximations.

scholarly journals Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

2020 ◽  
pp. 56-71
Author(s):  
Jenica Cringanu
Keyword(s):  
Banach Space ◽  
Variational Method ◽  
Critical Points ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Existence Results ◽  
Duality Mapping ◽  
Topological Methods ◽  
Carathéodory Function ◽  
Direct Variational Method

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

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 Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations

2018 ◽  
Vol 23 (4) ◽  
pp. 475-492 ◽  
Author(s):  
Jianxin He ◽  
Xinguang Zhang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Growth Conditions ◽  
Radial Solutions ◽  
Local Growth ◽  
Hessian Equations ◽  
Existence And Nonexistence ◽  
The Fixed Point Theorem

In this paper, we establish the existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations in a ball. Under some suitable local growth conditions for nonlinearity, several new results are obtained by using the fixed-point theorem.

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.

Diffraction from arrays of antiphase boundaries in ordered III-V alloys

1987 ◽  
Vol 45 ◽  
pp. 312-313
Author(s):  
T. S. Kuan
Keyword(s):  
High Surface ◽  
Growth Conditions ◽  
Ternary Alloys ◽  
Local Growth ◽  
Antiphase Boundaries ◽  
Surface Mobility ◽  
Ordered Phases ◽  
Diffraction Patterns ◽  
Atomically Flat ◽  
Degree Of Order

Recent electron diffraction studies have found ordered phases in AlxGa1-xAs, GaAsxSb1-x, and InxGa1-xAs alloy systems, and these ordered phases are likely to be found in many other III-V ternary alloys as well. The presence of ordered phases in these alloys was detected in the diffraction patterns through the appearance of superstructure reflections between the Bragg peaks (Fig. 1). The ordered phase observed in the AlxGa1-xAs and InxGa1-xAs systems is of the CuAu-I type, whereas in GaAsxSb1-x this phase and a chalcopyrite type ordered phase can be present simultaneously. The degree of order in these alloys is strongly dependent on the growth conditions, and during the growth of these alloys, high surface mobility of the depositing species is essential for the onset of ordering. Thus, the growth on atomically flat (110) surfaces usually produces much stronger ordering than the growth on (100) surfaces. The degree of order is also affected by the presence of antiphase boundaries (APBs) in the ordered phase. As shown in Fig. 2(a), a perfectly ordered In0.5Ga0.5As structure grown along the <110> direction consists of alternating InAs and GaAs monolayers, but due to local growth fluctuations, two types of APBs can occur: one involves two consecutive InAs monolayers and the other involves two consecutive GaAs monolayers.

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