On a Nonlocal Fractional p(.,.)-Laplacian Problem with Competing Nonlinearities

The aim of this paper is to study the existence of nontrivial weak solutions for the problem {M(integral Omega x Omega vertical bar u(x)-u(y)vertical bar(p(x,y))/p(x,y)vertical bar x-y vertical bar(N)+p(x,y)s dxdy) (Delta)(p(x,.))(s) u(x) = lambda f (x, u) - vertical bar u(x)vertical bar(q(x)-2)u(x) in Omega, u = 0 in partial derivative Omega, where Omega subset of R-N, N >= 2 is a bounded smooth domain, M and f are two continuous functions and (Delta)(p(.,.))(s) is the fractional p(.,.)-Laplacian while lambda is a positive parameter and 0 < s < 1. Using variational techniques combined with the theory of the generalized Lebesgue Sobolev spaces, we prove some existence and multiplicity results for the problem in an appropriate space of functions.