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 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 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 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.

The spectrum of the mean curvature operator

2020 ◽  
pp. 1-13
Author(s):  
Mihai Mihăilescu
Keyword(s):  
Boundary Condition ◽  
Mean Curvature ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Curvature Operator ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Principal Frequency ◽  
Mean Curvature Operator ◽  
The Mean ◽  
The Laplace Operator

Abstract We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝ N (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

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.

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