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

We 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.

Existence, Nonexistence and Multiplicity Results for Positive Radial Solutions of n−dimensional Elliptic System

Aims/ Objectives: In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the n−dimensional elliptic system systems have been widely studied, but there is relatively little research on n-dimensional elliptic systems. We are very interested in this subject and want to study it. We give new conclusions on the existence, nonexistence and multiplicity of positive solutions for the n-dimensional elliptic system. Study Design: Study on the existence, nonexistence and multiplicity of positive solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science & Technology University, September 2019 to present. Methodology: We prove the existence, nonexistence and multiplicity of positive solutions by the results of fixed point index. Results: We give new conclusions of existence, nonexistence and multiplicity of positive solutionsfor the system. Conclusion: We prove the existence, nonexistence and multiplicity of positive solutions to the n-dimensional elliptic system and give new conclusions.

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

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).

Positive radial solutions for a noncooperative resonant nuclear reactor model with sign-changing nonlinearities

This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system: $$ \textstyle\begin{cases} -\Delta u=uv+f( \vert x \vert ,u), & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ -\Delta v=cg(u)-dv, & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ \frac{\partial u}{\partial \textbf{n}}=0= \frac{\partial v}{\partial \textbf{n}},& \vert x \vert =R_{1}, \vert x \vert =R_{2}, \end{cases} $$ { − Δ u = u v + f ( | x | , u ) , 0 < R 1 < | x | < R 2 , x ∈ R N , − Δ v = c g ( u ) − d v , 0 < R 1 < | x | < R 2 , x ∈ R N , ∂ u ∂ n = 0 = ∂ v ∂ n , | x | = R 1 , | x | = R 2 , where $\mathbb{R}^{N}$ R N ($N\geq 1$ N ≥ 1 ) is the usual Euclidean space, n indicates the outward unit normal vector, $f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})$ f ∈ C ( [ R 1 , R 2 ] × [ 0 , ∞ ) , R ) , $g\in C([0,\infty ),[0,\infty ))$ g ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) , and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature.