Multiple results for critical quasilinear elliptic systems involving concave-convex nonlinearities and sign-changing weight functions∗

This paper is devoted to study the multiplicity of nontrivial nonnegative or positive solutions to the following systems    −4pu = λa1(x)|u|q−2u + b(x)Fu(u, v), in Ω, −4pv = λa2(x)|v|q−2v + b(x)Fv(u, v), in Ω, u = v = 0, on ∂Ω, where Ω ⊂ R is a bounded domain with smooth boundary ∂Ω; 1 < q < p < N , p∗ = Np N−p ; 4pw = div(|∇w|p−2∇w) denotes the p-Laplacian operator; λ > 0 is a positive parameter; ai ∈ L(Ω)(i = 1, 2) with Θ = p∗ p∗−q and b ∈ L∞(Ω) are allowed to change sign; F ∈ C((R), R) is positively homogeneous of degree p∗, that is, F (tz) = t ∗ F (z) holds for all z ∈ (R) and t > 0, here, R = [0, +∞). The multiple results of weak solutions for the above critical quasilinear elliptic systems are obtained by using the Ekeland’s variational principle and the mountain pass theorem.