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

Abstract

Let E = {u ∈ C1[0, 1]: u(0) = u(1) = 0}. Let Skv with v = {+, −} denote the set of functions u ∈ E which have exactly k − 1 interior nodal zeros in (0, 1) and vu be positive near 0. We show the existence of S-shaped connected component of Skv-solutions of the problem {(u′1−u′2)′+λa(x)f(u)=0,x∈(0,1),u(0)=u(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)}^\prime } + \lambda a(x)f(u) = 0,}&{x \in (0,1)} \end{array}} \\ {u(0) = u(1) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \right.$$\end{document} where λ > 0 is a parameter, a ∈ C([0, 1], (0, ∞)). We determine the intervals of parameter λ in which the above problem has one, two or three Skv-solutions. The proofs of the main results are based upon the bifurcation technique.