Variable order nonlocal Choquard problem with variable exponents

In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents (-Delta)(p(.))(s(.))u(x) = lambda vertical bar u(x)vertical bar(alpha(x)-2u)(x) +(integral(Omega) F(y, u(y))/vertical bar x - y vertical bar(u(x,y)) dy) f(x, u(x)), x is an element of Omega, u(x) = 0, x is an element of Omega(epsilon) := R-N\Omega, where Omega subset of R-N is a smooth and bounded domain, N >= 2, p, s, mu and alpha are continuous functions on R-N x R-N and f(x, t) is a Caratheodory function with F(x, t) := integral(t)(0) f(x, s) ds. Under suitable assumption on s, p, mu, alpha and f(x, t), first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.