Quasilinear elliptic problems via nonlinear Rayleigh quotient

Abstract

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