Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 321-333, 2023
$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator
Ruyun Ma, Zhiqian He, Xiaoxiao Su
Received January 20, 2020. Published online March 8, 2023.
Abstract: Let $E=\{u\in C^1[0,1] \colon u(0)=u(1)=0\}$. Let $S_k^\nu$ with $\nu=\{+, -\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^\nu$-solutions of the problem $\begin{cases}\biggl(\frac{u'}{\sqrt{1-u'^2}}\bigg)^{\prime}+\lambda a(x) f(u)=0, & x\in(0,1),\\ u(0)=u(1)=0, & \end{cases}$ where $\lambda>0$ is a parameter, $a\in C([0, 1], (0,\infty))$. We determine the intervals of parameter $\lambda$ in which the above problem has one, two or three $S_k^\nu$-solutions. The proofs of the main results are based upon the bifurcation technique.
Keywords: mean curvature operator; $S_k^\nu$-solution; bifurcation; Sturm-type comparison theorem
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Affiliations: Ruyun Ma (corresponding author), Department of Mathematics, Northwest Normal University, 967 Anning E. Road, Lanzhou 730070, P. R. China, e-mail: mary@nwnu.edu.cn; Zhiqian He, Department of Mathematics, Northwest Normal University, 967 Anning E. Road, Lanzhou 730070, P. R. China and School of Mathematics and Physics, Qinghai University, Xining 810016, P. R. China, e-mail: zhiqianhe1987@163.com; Xiaoxiao Su, School of Mathematics and Statistics, Xidian University, No. 2 South Taibai Road, Xi'an, Shaanxi 710071, P. R. China, e-mail: suxiaoxiao2856@163.com
