On the eigencurves of one dimensional p-Laplacian with weights for an elliptic Neumann problem

In the present paper we study the existence of the eigencurves of one dimensional p-Laplacian with indefinite weights of the Neumann problem for the following elliptic equation -|u′|p-2u′′=αm1(x)+βm2(x)|u|p-2uinI=]a,b[u′(a)=u′(b)=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\left( |u'|^{p-2}u'\right) ' =\left( \alpha \, m_{1}(x)+\beta \,m_{2}(x)\right) |u|^{p-2}u\quad \text{ in } \; I = {]a,b[}\\ \quad \\ u'(a)=u'(b) = 0. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{array} \right. \end{aligned}$$\end{document}Moreover, we establish their variational formulations and asymptotic behavior. Finally, we prove some properties of the principal eigencurve such as concavity and differentiability.