Positive Solution of Discrete BVPs Involving the Mean Curvature Operator on the Set of Nonnegative Integers

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Liping Wei ◽  
Ruyun Ma ◽  
Zhongzi Zhao
Keyword(s):  
Positive Solution ◽  
Mean Curvature ◽  
Curvature Operator ◽  
Mean Curvature Operator ◽  
The Mean

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.

Positive radial solutions for Dirichlet problems involving the mean curvature operator in Minkowski space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhiqian He ◽  
Liangying Miao
Keyword(s):  
Mean Curvature ◽  
Fixed Point Index ◽  
Curvature Operator ◽  
Positive Parameter ◽  
Point Index ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Positive Radial Solutions ◽  
Mean Curvature Operator ◽  
The Mean

Abstract In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type (P) − d i v ∇ u 1 − | ∇ u | 2 = λ f ( u )   in B 1 , u = 0                     on ∂ B 1 , $$\left\{\begin{aligned}\hfill & -\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u}{\sqrt{1-\vert \nabla u{\vert }^{2}}}\right)=\lambda f(u)\quad \text{in}\enspace {B}_{1},\hfill \\ \hfill & u=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \enspace \text{on}\enspace \partial {B}_{1},\enspace \hfill \end{aligned}\right.$$ where λ is a positive parameter, B 1 = { x ∈ R N : | x | < 1 } ${B}_{1}=\left\{x\in {\mathbb{R}}^{N}:\vert x\vert {< }1\right\}$ , f : [0, ∞) → [0, ∞) is a continuous function. Using the fixed point index in a cone, we prove the results on both uniqueness and multiplicity of positive radial solutions of (P).

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.

A weighted quasilinear equation related to the mean curvature operator

2012 ◽  
Vol 75 (15) ◽  
pp. 5905-5923 ◽  
Author(s):  
Santiago Cano-Casanova ◽  
Julián López-Gómez ◽  
Kazuhiro Takimoto
Keyword(s):  
Mean Curvature ◽  
Quasilinear Equation ◽  
Curvature Operator ◽  
Mean Curvature Operator ◽  
The Mean
Sign in / Sign up

Export Citation Format

Share Document