Existence of solutions to Kirchhoff type equations involving the nonlocal p1&...&pm fractional Laplacian with critical Sobolev-Hardy exponent

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p(1)&... &p(m) fractional Laplacian with critical Sobolev-Hardy exponent {M-1 (integral(R2N) |u(x) - u(y)|(p1)/|x - y|(N+p1s1) dx dy) (-Delta)(p1)(s1)u+ center dot center dot center dot +M-m (integral R-2N |u(x) - u(y)|(pm)/|x - y|(N+pmsm) dx dy) (-Delta)(pm)(sm) u = f (x, u) +gamma |u|(ps)*((alpha)-2)u/|x|(alpha) + beta |u(|q-2)u/|x|(alpha) in Omega, u = 0 in R-N \ Omega, where 0< s(m) < center dot center dot center dot < s(1) = s < 1, 1 < p(m) <= center dot center dot center dot <= p(1) = p < N/s, m >= 1, beta, gamma are nonnegative constants and p(s)*(alpha) = p(N-alpha)/N-sp <= p(s)* (0) is called the critical Sobolev-Hardy exponent, 1 < q < p, 0 <= alpha < ps. Here (-Delta)(r)(s), with r is an element of {p(1), ... , p(m)} is the fractional r-Laplace operator. Omega is an open bounded subset of R-N with smooth boundary and 0 is an element of Omega. M-1, ... , M-m are continuous functions and f is a Caratheodory function which does not satisfy the AmbrosettiRabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when the gamma = 0. We also study the existence of two nontrivial solutions for Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, equation we consider the case N = ps, and study a degenerate Kirchhoff involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.