This paper examines a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions. With the aid of the Nehari manifold, the existence of multiple nontrivial nonnegative solutions is obtained. 2010 AMS Mathematics subject classification. Primary 35A15, 35J60, 35S15.Thus,there exists a t b,max > 0 such that,then there is a unique 0 < t + < t b,max such that t + u ∈ N + λ and J λ (t + u) = inf t≥0 J λ (tu);

This paper is devoted to study a class of fractional equations with critical exponent, concave nonlinearity and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.Mathematics Subject Classification. 35J20, 35S15, 47J30, 47G20.

Abstract. Three nontrivial nonnegative solutions for some critical quasilinear elliptic systems with lower-order negative perturbations are obtained by using the Ekeland's variational principle and the mountain pass theorem.

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