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The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocalfractional Laplacian with critical Sobolev-Hardy exponentwhereare nonnegative constants andis called the critical Sobolev-Hardy exponent,. Herewithis the fractional  r -Laplace operator. Ω is an open bounded subset ofwith smooth boundary and.are continuous functions and  f  is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz 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. We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional  p -Laplacian via Morse theory. Finally, we consider the caseand study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.

complex variables and elliptic equations

Existence of solutions to Kirchhoff type equations involving the nonlocal  p                  1 &amp; … &amp; p                                      m                   fractional Laplacian with critical Sobolev-Hardy exponent