The Nehari manifold for fractional Kirchhoff problems involving singular and critical terms

In the present paper, we study the following singular Kirchhoff problem {M(integral integral(R2N) vertical bar u(x) - u(y)vertical bar(2)/vertical bar x - y vertical bar(N+2s)dxdy) (-Delta)(s)u =lambda f(x)u(-gamma) + g(x)u(2s)*(-1) in Omega, u > 0 in Omega, u = 0 in R-N\Omega, where Omega subset of R-N is an open bounded domain, dimension N > 2s with s is an element of(0, 1), 2(s)* = 2N/(N - 2s) is the fractional critical Sobolev exponent, parameter lambda > 0, exponent gamma is an element of(0, 1), M models a Kirchhoff coefficient, f is an element of L-2s*(/2s)*(+gamma-1) (Omega) is a positive weight, while g is an element of L-infinity(Omega) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of lambda. This approach allows us to avoid any restriction on the boundary of Omega. (C) 2018 Elsevier Ltd. All rights reserved.