Quasilinear elliptic problems via nonlinear Rayleigh quotient

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$\{\begin{array}{lrr}-{\mathrm{\Delta }}_{\mathrm{\Phi }}u=\lambda a(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,& x\in \mathrm{\Omega },u=0,& x\in \mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N,N\ge 2,$ is a smooth bounded domain, $1<q<\mathrm{\ell }\le m<p<{\mathrm{\ell }}^{\ast }$ and $\mathrm{\Phi }:\mathbb{R}\to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number ${\lambda }^{\ast }>0$ in such way that our main problem admits at least two distinct solutions for each $\lambda \in (0,{\lambda }^{\ast })$. Furthermore, using some fine estimates and some extra assumptions on $\mathrm{\Phi }$, we prove the existence of at least two positive solutions for $\lambda ={\lambda }^{\ast }$ and $\lambda \in ({\lambda }^{\ast },\bar{\lambda })$ where $\bar{\lambda }>{\lambda }^{\ast }$.