scholarly journals Infinitely Many Solutions for Discrete Boundary Value Problems with the p , q -Laplacian Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zhuomin Zhang ◽  
Zhan Zhou
Keyword(s):  
Maximum Principle ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Point Theory ◽  
Strong Maximum Principle ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Existence And Multiplicity

In this paper, we consider the existence and multiplicity of solutions for a discrete Dirichlet boundary value problem involving the p , q -Laplacian. By using the critical point theory, we obtain the existence of infinitely many solutions under some suitable assumptions on the nonlinear term. Also, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions.

scholarly journals Existence results for a sublinear second order Dirichlet boundary value problem on the half-line

2020 ◽  
Vol 40 (5) ◽  
pp. 537-548
Author(s):  
Dahmane Bouafia ◽  
Toufik Moussaoui
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Half Line ◽  
Dirichlet Boundary Value Problem ◽  
Nontrivial Solutions

In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by using Ekeland's variational principle and critical point theory.

scholarly journals Three solutions for a partial discrete Dirichlet boundary value problem with p-Laplacian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shaohong Wang ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Point Theory ◽  
Strong Maximum Principle ◽  
Dirichlet Boundary Value Problem ◽  
Partial Difference Equation ◽  
Three Solutions ◽  
Partial Difference

AbstractBy employing critical point theory, we investigate the existence of solutions to a boundary value problem for a p-Laplacian partial difference equation depending on a real parameter. To be specific, we give precise estimates of the parameter to guarantee that the considered problem possesses at least three solutions. Furthermore, based on a strong maximum principle, we show that two of the obtained solutions are positive under some suitable assumptions of the nonlinearity.

scholarly journals Small Solutions of the Perturbed Nonlinear Partial Discrete Dirichlet Boundary Value Problems with (p,q)-Laplacian Operator

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1207
Author(s):  
Feng Xiong ◽  
Zhan Zhou
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Critical Point ◽  
Critical Point Theory ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Laplacian Operator

In this paper, we consider a perturbed partial discrete Dirichlet problem with the (p,q)-Laplacian operator. Using critical point theory, we study the existence of infinitely many small solutions of boundary value problems. Without imposing the symmetry at the origin on the nonlinear term f, we obtain the sufficient conditions for the existence of infinitely many small solutions. As far as we know, this is the study of perturbed partial discrete boundary value problems. Finally, the results are exemplified by an example.

scholarly journals Multiple Solutions for Partial Discrete Dirichlet Problems Involving the p-Laplacian

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2030
Author(s):  
Sijia Du ◽  
Zhan Zhou
Keyword(s):  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Discrete Variables ◽  
Dirichlet Problems ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
The Maximum Principle

Due to the applications in many fields, there is great interest in studying partial difference equations involving functions with two or more discrete variables. In this paper, we deal with the existence of infinitely many solutions for a partial discrete Dirichlet boundary value problem with the p-Laplacian by using critical point theory. Moreover, under appropriate assumptions on the nonlinear term, we determine open intervals of the parameter such that at least two positive solutions and an unbounded sequence of positive solutions are obtained by using the maximum principle. We also show two examples to illustrate our results.

scholarly journals Existence of Solution to a Second-Order Boundary Value Problem via Noncompactness Measures

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Wen-Xue Zhou ◽  
Jigen Peng
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Measure Of Noncompactness ◽  
Dirichlet Boundary ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Of Solution ◽  
Dirichlet Boundary Value Problem ◽  
Condensing Mapping

The existence and uniqueness of the solutions to the Dirichlet boundary value problem in the Banach spaces is discussed by using the fixed point theory of condensing mapping, doing precise computation of measure of noncompactness, and calculating the spectral radius of linear operator.

scholarly journals Multiple Solutions for Boundary Value Problems of p-Laplacian Difference Equations Containing Both Advance and Retardation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhenguo Wang ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Open Interval ◽  
Point Theory ◽  
Infinitely Many Solutions

This paper concerns the existence of solutions for the Dirichlet boundary value problems of p-Laplacian difference equations containing both advance and retardation depending on a parameter λ. Under some suitable assumptions, infinitely many solutions are obtained when λ lies in a given open interval. The approach is based on the critical point theory.

scholarly journals Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 381 ◽  
Author(s):  
Jianxia Wang ◽  
Zhan Zhou
Keyword(s):  
Mean Curvature ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Point Theory ◽  
Curvature Operator ◽  
Constant Sign ◽  
Dirichlet Boundary Value Problem ◽  
Constant Sign Solutions ◽  
Mean Curvature Operator ◽  
Truncation Techniques

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

Existence and Multiplicity of Positive Solutions for a Dirichlet Boundary Value Problem in ℝ2

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
V. Barutello ◽  
A. Capietto ◽  
P. Habets
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Degree Theory ◽  
Dirichlet Boundary Value Problem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Parameter Dependent ◽  
Vector Differential Equation

AbstractWe deal with the Dirichlet boundary value problem associated to a parameter-dependent second order vector differential equation. Using the method of lower and upper solutions together with degree theory, we provide existence and multiplicity of positive solutions.

Existence and Multiple Solutions for Higher Order Difference Dirichlet Boundary Value Problems

2018 ◽  
Vol 19 (5) ◽  
pp. 539-544
Author(s):  
Lianwu Yang
Keyword(s):  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Point Theory ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Linking Theorem ◽  
Order Difference ◽  
Existence And Multiplicity

AbstractIn this paper, a higher order nonlinear difference equation is considered. By using the critical point theory, we obtain the existence and multiplicity for solutions of difference Dirichlet boundary value problems and give some new results. The proof is based on the variational methods and linking theorem.

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