Positive radial solutions for multiparameter Dirichlet systems with mean curvature operator in Minkowski space and Lane–Emden type nonlinearities

2019 ◽  
Vol 266 (9) ◽  
pp. 5377-5396 ◽  
Author(s):  
Daniela Gurban ◽  
Petru Jebelean
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Curvature Operator ◽  
Radial Solutions ◽  
Positive Radial Solutions ◽  
Mean Curvature Operator

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 On a power-type coupled system with mean curvature operator in Minkowski space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhiqian He ◽  
Yanzhong Zhao ◽  
Liangying Miao
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Index Theory ◽  
Coupled System ◽  
Curvature Operator ◽  
Radial Solutions ◽  
Uniqueness Results ◽  
Positive Radial Solutions ◽  
Completely Continuous Operators ◽  
Continuous Operators

AbstractWe study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$ { M ( u ) + v α = 0 in  B , M ( v ) + u β = 0 in  B , u | ∂ B = v | ∂ B = 0 , where $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) and B is a unit ball in $\mathbb{R}^{N} (N\geq 2)$ R N ( N ≥ 2 ) . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

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.

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