$S$-shaped connected component for a nonlinear Dirichlet problem involving mean curvature operator in one-dimension Minkowski space
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1891-1908
https://doi.org/10.4134/BKMS.b180011
Published online November 1, 2018
Ruyun Ma, Man Xu
Northwest Normal University, Northwest Normal University
Abstract : In this paper, we investigate the existence of an $S$-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$ \left\{ \begin{align*}&\Big(\frac{u'}{\sqrt{1-u'^2}}\Big)'+\lambda a(x)f(u)=0,\ \ \ x\in(0,1),\\ &u(0)=u(1)=0,\\\end{align*} \right.$$ where $\lambda$ is a positive parameter, $f\in C[0,\infty)$, $a\in C[0,1]$. The proofs of main results are based upon the bifurcation techniques.
Keywords : $S$-shaped connected component, positive solutions, mean curvature operator, Minkowski space, bifurcation
MSC numbers : 34B10, 34B18, 34C23, 35B40, 35J65
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