Abstract
The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal fractional Laplacian with critical Sobolev-Hardy exponent
where
are nonnegative constants and
is called the critical Sobolev-Hardy exponent,
. Here
with
is the fractional r-Laplace operator. Ω is an open bounded subset of
with 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 case
and 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.
AMS Subject Classifications:
Acknowledgements
The authors wish to thank the editorial board and referees for a very careful reading of the manuscript, and for pointing out misprints that led to the improvement of the original manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).