On Critical Schrodinger-Kirchhoff-Type Problems Involving the Fractional p-Laplacian with Potential Vanishing at Infinity

The aim of this paper is to study the existence of solutions for critical Schrodinger-Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: M(integral integral(R2N) vertical bar u(x) - u(y)vertical bar(p)/vertical bar x - y vertical bar(N+sp) dxdy + integral(RN) V(x)vertical bar u(x)vertical bar(p) dx) ((-Delta)(p)(s)u(x) + V (x)vertical bar u vertical bar(p-2) u) = K(x)(lambda f(x, u) + vertical bar u vertical bar(p)*(s-2) u), where M : [0,infinity) -> [0,infinity) is a continuous function, (-Delta)(p)(s) is the fractional p-Laplacian, 0 < s < 1 < p < infinity with sp < N, p(s)* = Np/(N - ps), K, V are nonnegative continuous functions satisfying some conditions, and integral is a continuous function on R-N x R satisfying the AmbrosettiRabinowitz-type condition, lambda > 0 is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into L-K(alpha)(R-N), alpha is an element of [p, p(s)*]. Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do ' O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).