Existence of solutions for Schrodinger-Kirchhoff systems involving the fractional p-Laplacian in R-N

摘要

The aim of this paper is to study the existence of weak solutions for Schrodinger-Kirchhoff systems involving the fractional p-Laplacian {M([(u, v)](K),(s),(p)(p))L(p)(s)u(x) + V(x)vertical bar u vertical bar(p-2) u = lambda H-u(x, u, v) + g(1)(x), M([(u, v)](K,s,p)(p))L(p)(s)v(x) + V(x)vertical bar v vertical bar(p-2) v = lambda H-v(x, u, v) + g(2)(x), in R-N, where lambda is a real positive parameter, M : [0,infinity) -> (0,infinity) and V : R-N -> (0,infinity) are continuous functions, g(1) and g(2) are perturbation terms, L-p(s) is a nonlocal fractional operator with singular kernel K : R-N {0} -> R+, 0 = 0 such that H-z(x, z)z - mu H(x, z) >= -rho vertical bar z vertical bar(p) - phi(x) for all x is an element of R-N and z is an element of R-2 with vertical bar z vertical bar >= r, where z = (u, v), H-z(x, z) = (H-u(x, u, v), H-v(x, u, v)), vertical bar z vertical bar = root u(2) + v(2), 0 is an element of[1, p(s)*/p), rho >= 0 and 0 <= phi is an element of L-1 (R-N). By using the Mountain Pass Theorem and Ekeland's variational principle, we obtain the existence of solutions to the above system. Furthermore, we also investigate the existence of solutions for a system of equations with the critical exponent and the Hardy potential. Finally, we study the case that V can vanish on a set of measure zero in RN and H satisfies the Ambrosetti-Rabinowitz condition.