On a systems involving fractional Kirchhoff‐type equations and Krasnoselskii's genus

Abstract

We consider a class of variational systems involving fractional Kirchhoff‐type equations of the form: M1false(false‖1ptu1pt‖X2false)false(−normalΔfalse)su=Fufalse(x,u,vfalse)1em1emin1em1emnormalΩ,M2false(false‖1ptv1pt‖X2false)false(−normalΔfalse)sv=Fvfalse(x,u,vfalse)1em1emin1em1emnormalΩ,u=v=0in1em1emRN\normalΩ, where s ∈ (0,1), N > 2s, normalΩ⊂RN a smooth and bounded domain, the functions Fu, Fv, M1 and M2 are continuous and ( − Δ)s is the fractional Laplacian operator. In this paper, we show that, under appropriate growth conditions on the nonlinearities Fu and Fv and on the nonnegative functions M1 and M2, the (weak) solutions are precisely the critical points of a related functional defined on a fractional Hilbert space Y(Ω) = X(Ω) × X(Ω) and the existence infinitely many solutions can be obtained by the use of the Krasnoselskii's genus. Besides, a regularity result can also be obtained by using specific results for systems in conjunction with the growth assumptions of these functions.