A variational approach to the boundary value problem for hypersurfaces with prescribed mean curvature

1984 ◽  
pp. 50-63
Author(s):  
Shôichirô Takakuwa
Keyword(s):  
Boundary Value Problem ◽  
Mean Curvature ◽  
Variational Approach ◽  
Boundary Value ◽  
Prescribed Mean Curvature

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

2018 ◽  
Vol 29 (10) ◽  
pp. 1850069
Author(s):  
Hong Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Boundary Value ◽  
Fundamental Solutions ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Manifold With Boundary ◽  
Prescribed Mean Curvature Equation

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.

scholarly journals A Variational Approach to Perturbed Discrete Anisotropic Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Amjad Salari ◽  
Giuseppe Caristi ◽  
David Barilla ◽  
Alfio Puglisi
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variational Approach ◽  
Boundary Value ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Multiplicity Results ◽  
Anisotropic Equation

We continue the study of discrete anisotropic equations and we will provide new multiplicity results of the solutions for a discrete anisotropic equation. We investigate the existence of infinitely many solutions for a perturbed discrete anisotropic boundary value problem. The approach is based on variational methods and critical point theory.

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 S-shaped connected component of radial positive solutions for a prescribed mean curvature problem in an annular domain

2019 ◽  
Vol 17 (1) ◽  
pp. 929-941
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Mean Curvature ◽  
Boundary Value ◽  
Normal Derivative ◽  
Euclidean Norm ◽  
Positive Parameter ◽  
Annular Domain ◽  
Connected Component ◽  
Curvature Problem

Abstract In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem $$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$ where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2}, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $ s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞), $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.

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