A Mathematic Expression of the Genes of Chinese Traditional Philosophy
We provide a mathematic model for the Traditional Yin-and-Yang Double Fish Diagram which from Chinese Traditional Philosophy.
math.HO↗
arXiv subjects
Ruyun Ma
Publications and source records attributed to Ruyun Ma.
Proof of the Bonheure-Noris-Weth conjecture on oscillatory radial solutions of Neumann problems
Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(β) = β, \ f(s) s\ \text{for}\ s\in (β, \infty)$ and $f'(β)>λ^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -Δu+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_νu=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $β$ provided that $f'(β) >λ^r_k$ for $k>2$. In this paper, we show that the answer is yes.
math.AP↗
On multiple positive ground state solutions for a mean curvature equation in Minkowski space
In this paper, we show how changes in the sign of nonlinearity leads to multiple radial ground state solutions of the mean curvature equation $ \nabla\cdot \Big[\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big] +λf(u)=0\ \ \text{in} \ \mathbb{R}^N $ for sufficiently large $λ$ with $N\geq 2$.
math.AP↗
Spectrum structure for eigenvalue problems involving mean curvature operators in Euclidean and Minkowski spaces
In this paper, we are concerned with quasilinear Dirichlet problem $$ \left\{ \aligned &-\Big(\frac{u'(x)}{\sqrt{1+κ(u'(x))^2}}\Big)'=λu(x), \ \ \ \ \ 0<x<1,\\ &u(0)= u(1)=0,\\ \endaligned \right. \eqno (P) $$ where $κ\in (-\infty, 0)\cup (0, \infty)$ is a constant. We show that any nontrivial solution $ u$ of (P) has only finite many of simple zeros in $[0,1]$, all of humps of $u$ are same, and the first hump is symmetric around the middle point of its domain. We also describe the global structure of the set of nontrivial solutions of (P).
math.CA↗
Global structure of radial sign-changing solutions for the prescribed mean curvature problem in a ball
In this paper, we are concerned with the global structure of radial solutions, with prescribed nodal properties, to the boundary value problem $$\text{div}\big(ϕ_{N}(\nabla v)\big)+λf(|x|, v)=0 ~~~\text{in} ~~B(R), ~~~ v=0 ~~~\text{on} ~~\partial B(R), $$ where $ϕ_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}},\; y\in \mathbb{R}^{N}$, $λ$ is a positive parameter, $B(R)=\{x\in \mathbb{R}^{N} :|x|<R\}$, and $|\cdot|$ denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the behavior of nonlinear term $f$ near 0, are obtained by using global bifurcation techniques.
math.AP↗
Global structure of radial positive solutions for a prescribed mean curvature problem in a ball
In this paper, we are concerned with the global structure of radial positive solutions of boundary value problem$$\text{div}\big(ϕ_{N}(\nabla v)\big)+λf(|x|, v)=0 \text{in} B(R), v=0 \text{on} \partial B(R), $$where $ϕ_{N}(y)=\frac{y}{\sqrt{1-|y|^{2}}}, y\in \mathbb{R}^{N}$, $λ$ is a positive parameter, $B(R)=\{x\in \mathbb{R}^{N} :|x|<R\}$, and $|\cdot|$ denote the Euclidean norm in $\mathbb{R}^{N}$. All results, depending on the behavior of nonlinear term $f$ near 0, are obtained by using global bifurcation techniques.
math.AP↗
Unilateral global bifurcation and nodal solutions for the $p$-Laplacian with sign-changing weight
In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν(p), 0)$, where $μ_k^ν(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in\{+,-\}$. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.
math.AP↗