Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth

The aim of this paper is to study a degenerate Kirchhoff-type elliptic problem driven by the fractional Laplace operator with variable order derivative and variable exponents. More precisely, we consider {[u](s(.))(2(theta-1)) (-Delta)(s(.)) u = lambda a(x)vertical bar u vertical bar p((x)-2)u + b(x)vertical bar u vertical bar q((x)-2)u in Omega, u = 0 in R-N \ Omega, where [u](s(.)) is the Garliado seminorm, theta > 1, N >= 1, s(.) : R-N x R-N -> (0, 1) is a continuous function,. lambda > 0 is a parameter, Omega is a bounded domain in R-N with N > 2s(x, y) for all (x, y) is an element of Omega x Omega, (-Delta)(s(.)) is the variable-order fractional Laplacian, a, b is an element of L-infinity(Omega) are two positive weight functions, and p, q is an element of C((Omega) over bar) with 1 < p(x) < 2 theta < q(x) < 2N/(N - 2s(x, x)). Under some suitable assumptions, we obtain that the problem admits at least two distinct nonnegative solutions by applying the Nehari manifold approach, provided lambda is sufficiently small. Moreover, the existence of infinitely many solutions is also investigated by the symmetric mountain pass theorem. (C) 2020 Elsevier Inc. All rights reserved.