Quasilinear elliptic problems via nonlinear rayleigh quotient

Abstract

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems:{−ΔΦu = 𝜆a(x)|u|q−2u + |u|p−2u, x ∈ Ω,u = 0, x ∈ 𝜕Ω,where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, 1 < q < 𝓁 ≤ m < p < 𝓁∗ and Φ : ℝ → 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∗ > 0in such a way that our main problem admits at least two distinct solutions for each 𝜆 ∈ (0, 𝜆∗ ). Furthermore, using somefine estimates and some extra assumptions on Φ, we prove the existence of at least two positive solutions for 𝜆 = 𝜆∗and 𝜆 ∈ (𝜆∗, 𝜆) where 𝜆 > 𝜆∗