Multiple solutions of systems involving fractional Kirchhoff-type equations with critical growth

Abstract

. In this paper we are going to study existence and multiplicity of solutions of a system involving fractional Kirchhoff-type and critical growth of form where s ∈ ( 0 , 1 ) , n > 2 s , Ω ⊂ R n is a bounded and open set, 2 ∗ s = 2 n / ( n − 2 s ) denotes the fractional critical Sobolev exponent, the functions M 1 , M 2 , f and g are continuous functions, ( − Δ ) s is the fractional laplacian operator, || . || X is a norm in the fractional Hilbert Sobolev space X ( Ω ) , F ( x , v ( x )) = v x , G x , ( x )) g ( τ ) d τ , r 1 and r 2 are positive constants, λ and γ are real parameters. For this problem we prove the existence of in fi nitely many solutions, via a suitable truncation argument and exploring the genus theory introduced by Krasnoselskii. Also we show that these solutions are suf fi ciently regular and solve the problem pointwise.