On a class of fractional p(x) -Kirchhoff type problems. Issue 2 (25th January 2021) - Azroul, E.; Benkirane, A.; Shimi, M.; Srati, M.
Abstract : This paper is concerned with a class of fractional p ( x ) -Kirchhoff type problems with Dirichlet boundary data of the following form ( P M s ) M ∫ Q 1 p ( x, y ) | u ( x ) − u ( y ) | p ( x, y ) | x − y | N + s p ( x, y ) d x d y − Δ p ( x ) s u ( x ) = λ | u ( x ) | r ( x ) − 2 u ( x )   in  Ω, u = 0   in  R N ∖ Ω . By means of mountain pass theorem of Ambrosetti and Rabinowitz, direct variational approach and Ekeland's variational principle, we investigate the existence of nontrivial weak solutions for the above problem in different cases of the competition between the growth rates of functions p and r involved in problem ( P M s ), this fact is essential in describing the set of eigenvalues of this problem.