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 {-?pu = ?a1(x)|u|q-2u+b(x)Fu(u,v), in ?, {-?pv = ?a2(x)|v|q-2v+b(x)Fv(u,v), in ?, { u = v = 0, on ??, where ? ? RN is a bounded domain with smooth boundary ??; 1 < q < p < N, p* = (Np)/(N - p); ?pw = div(|?w|p-2?w) denotes the p-Laplacian operator; ? > 0 is a positive parameter; ai ? L?(?)(i = 1, 2) with ? = p*/ (p* - q) and b ? L?(?) are allowed to change sign; F ? C1((R+)2,R+) is positively homogeneous of degree p*, that is, F(tz) = t p*F(z) holds for all z ? (R+)2 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.