scholarly journals Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with ϕc-Laplacian

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1839 ◽  
Author(s):  
Yanshan Chen ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Curvature Operator ◽  
Nontrivial Solutions ◽  
Three Solutions ◽  
The Maximum Principle ◽  
Mean Curvature Operator ◽  
The Mean

In this paper, based on critical point theory, we mainly focus on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator. Without imposing the symmetry or oscillating behavior at infinity on the nonlinear term f, we respectively obtain the sufficient conditions for the existence of at least three non-trivial solutions and the existence of at least two non-trivial solutions under different assumptions on f. In addition, by using the maximum principle, we also deduce the existence of at least three positive solutions from our conclusion. As far as we know, our results are supplements to some well-known ones.

scholarly journals Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

2019 ◽  
Vol 17 (1) ◽  
pp. 1055-1064 ◽  
Author(s):  
Jiaoxiu Ling ◽  
Zhan Zhou
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Mean Curvature ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Curvature Operator ◽  
Mean Curvature Operator ◽  
The Mean

Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space

2017 ◽  
Vol 17 (4) ◽  
pp. 769-780 ◽  
Author(s):  
Daniela Gurban ◽  
Petru Jebelean ◽  
Călin Şerban
Keyword(s):  
Minkowski Space ◽  
Critical Point Theory ◽  
Mean Curvature ◽  
Lower Semicontinuous ◽  
Point Theory ◽  
Curvature Operator ◽  
Nontrivial Solutions ◽  
Potential System ◽  
Mean Curvature Operator ◽  
The Mean

AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

scholarly journals On the existence of multiple solutions for a partial discrete Dirichlet boundary value problem with mean curvature operator

2021 ◽  
Vol 11 (1) ◽  
pp. 198-211
Author(s):  
Sijia Du ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Mean Curvature ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Curvature Operator ◽  
Dirichlet Boundary Value Problem ◽  
The Maximum Principle ◽  
Mean Curvature Operator

Abstract Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.

scholarly journals Three Solutions for a Partial Discrete Dirichlet Problem Involving the Mean Curvature Operator

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1691
Author(s):  
Shaohong Wang ◽  
Zhan Zhou
Keyword(s):  
Mean Curvature ◽  
Dirichlet Boundary ◽  
Point Theory ◽  
Curvature Operator ◽  
Dirichlet Boundary Value Problem ◽  
Three Solutions ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Mean Curvature Operator ◽  
The Mean

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

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.

Existence and nonexistence results for a fourth-order discrete neumann boundary value problem

2014 ◽  
Vol 51 (2) ◽  
pp. 186-200
Author(s):  
Xia Liu ◽  
Yuanbiao Zhang ◽  
Haiping Shi
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Point Theory ◽  
Nonlinear Difference Equation ◽  
Neumann Boundary Value Problem ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

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.

scholarly journals Existence of three solutions to the discrete fourth-order boundary value problem with four parameters

2018 ◽  
Vol 38 (2) ◽  
pp. 173-185 ◽  
Author(s):  
Mohamed Ousbika ◽  
Zakaria El Allali
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Critical Point ◽  
Fourth Order ◽  
Critical Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Three Solutions

In this work, we willproving the existence of three solutionsf or the discrete nonlinear fourth order boundary value problems with four parameters. The methods used here are based on the critical point theory.

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