arXiv subjects

Hongliang Gao

Publications and source records attributed to Hongliang Gao.

3 recordsLinked to original sources

arXiv2014

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↗

arXiv2014

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↗

arXiv2014

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↗