$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator

Ma, Ruyun ; He, Zhiqian ; Su, Xiaoxiao

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 321-333.

Résumé

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 $$ \begin{cases} \biggl (\dfrac {u'}{\sqrt {1-u'^2}}\bigg )^{\prime }+\lambda a(x) f(u)=0, x\in (0,1), \\ u(0)=u(1)=0, \end{cases} $$ 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.

Détail

DOI : 10.21136/CMJ.2023.0027-20

Classification : 34C10, 34C23, 35B40, 35J65
Keywords: mean curvature operator; $S_k^\nu $-solution; bifurcation; Sturm-type comparison theorem