arXiv subjects

Tianlan Chen

Publications and source records attributed to Tianlan Chen.

3 recordsLinked to original sources

arXiv2015

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↗

arXiv2015

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↗

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↗