Global structure of positive solutions for p-Laplacian Neumann problem with indefinite weight

We are concerned with the global structure of positive solutions for p-Laplacian Neumann problem: where \(\varphi _p(s)=\vert s\vert ^{p-2}s\) ${\phi }_p(s)={|s|}^{p-2}s$ , \(p>1\) $p>1$ , \(\lambda >0\) $\lambda >0$ is a parameter,  \(h:[0,1]\rightarrow \mathbb {R}\) $h:[0,1]\to \mathbb{R}$ is a continuous function with \(\int _0^1h(x)\text {d}x<0\) ${\int }_0^{1}h(x)\text{d}x<0$ ,  \(g:[0,\infty )\rightarrow [0,\infty )\) $g:[0,\infty )\to [0,\infty )$ is a continuous function satisfying \(\lim _{s\rightarrow 0}g(s)/\varphi _p(s)=0\) ${lim}_{s\to 0}g(s)/{\phi }_p(s)=0$ and \(\lim _{s\rightarrow \infty }g(s)/\varphi _p(s)=0\) ${lim}_{s\to \infty }g(s)/{\phi }_p(s)=0$ . We obtain a \(\subset \) $\subset$ -shaped component of positive solutions of problem (P) provided suitable conditions. That is, there exist \(\lambda ^*>\lambda _*>0\) ${\lambda }^{\ast }>{\lambda }_{\ast }>0$ , such that the problem (P) has two positive solutions for \(\lambda >\lambda ^*\) $\lambda >{\lambda }^{\ast }$ and no positive solution for \(\lambda <\lambda _*\) $\lambda <{\lambda }_{\ast }$ . The proof of main result is based upon bifurcation technology. In addition, to prove the main result, we investigate the principal eigenvalue of auxiliary problem: \(\begin{aligned} \left\{ \begin{array}{l} -(\varphi _p(u'))'+\frac{1}{m}\varphi _p(u)=\lambda h(x) \varphi _p(u),\ \ x\in (0,1),\\ u'(0)=u'(1)=0, \ \ \\ \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-{({\phi }_p(u^{\prime }))}^{\prime }+\frac{1}{m}{\phi }_p(u)=\lambda h(x){\phi }_p(u),x\in (0,1),\hfill \\ u^{\prime }(0)=u^{\prime }(1)=0,\hfill \end{array}\right)\end{array}$$ where \(m\in \mathbb {N}^+\) $m\in {\mathbb{N}}^{+}$ .