Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Abstract

Let Ω∋0 be an-open bounded domain in ℝ𝑁(𝑁≥3) and 𝑝∗=(𝑝𝑁/(𝑁−𝑝)). We consider the following quasilinear elliptic system of two equations in 𝑊01,𝑝(Ω)×𝑊01,𝑝(Ω): −Δ𝑝𝑢=𝜆𝑓(𝑥)|𝑢|𝑞−2𝑢+(𝛼/(𝛼+𝛽))ℎ(𝑥)|𝑢|𝛼−2𝑢|𝑣|𝛽,−Δ𝑝𝑣=𝜇𝑔(𝑥)|𝑣|𝑞−2𝑣+(𝛽/(𝛼+𝛽))ℎ(𝑥)|𝑢|𝛼|𝑣|𝛽−2𝑣, where 𝜆,𝜇>0, Δ𝑝 denotes the 𝑝-Laplacian operator, 1≤𝑞1 satisfy 𝑝<𝛼+𝛽≤𝑝∗, and 𝑓,𝑔,ℎ are continuous functions on Ω which are somewhere positive but which may change sign on Ω. We establish the existence and multiplicity results of positive solutions to (the above mentioned quasilinear elliptic system equations) by variational methods.