The existence of multiple solutions of p-laplacian elliptic equation

Abstract

This paper considers the quasilinear elliptic equation (1.1)λ{-Δpu=|u|m-1u+λ|u|q-1u, x∈Ω,u∈W01,p(Ω),where -Δpu=-div(|Δu|p-2Δu), and 0<m<p-1<q<+∞, ω is a bounded domain in RN(N≥3). λ is a positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ lias at least one positive solution if λ υ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ υ (0,λ*) and q≤NpN-p-1. Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0<m<p-1<q=NpN-p-1.