Quasilinear elliptic problems via nonlinear Rayleigh quotient

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