Positive Solutions for BVPs with One-dimensional Mean Curvature Operator

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
João Marcos do Ó ◽  
Aleksandra Orpel
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Mean Curvature ◽  
Curvature Operator ◽  
Dirichlet Problems ◽  
One Dimensional ◽  
Functional Parameters ◽  
Mean Curvature Operator

AbstractWe discuss the existence and properties of solutions for systems of Dirichlet problems involving one dimensional mean curvature operator. Our approach is based on variational methods and covers both sublinear and superlinear cases of nonlinearities. We also investigate the continuous (in some sense) dependence of solutions on functional parameters.

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.

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.

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 Nonhomogeneous Nonlinear Dirichlet Problems with ap-Superlinear Reaction

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Mean Curvature ◽  
Differential Operators ◽  
Critical Groups ◽  
Curvature Operator ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Nonhomogeneous Differential Operator ◽  
Generalized Mean Curvature ◽  
Mean Curvature Operator ◽  
Superlinear Reaction

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reactionf(z,ζ), whose primitivef(z,ζ)isp-superlinear near±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like thep-Laplacian, the(p,q)-Laplacian, and thep-generalized mean curvature operator.

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