The critical Choquard equations with a Kirchhoff type perturbation in bounded domains

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains,{-(1+b parallel to u parallel to(2))Delta u=(integral(Omega)u(2)(y)/|x-y|(4)dy)u+lambda u in Omega, u=0 on partial derivative Omega,where Omega R-N(N >= 5) is a smooth bounded domain and parallel to & sdot;parallel to is the standard norm of H-0(1)(Omega). Under the suitable assumptions on the constant b >= 0, we prove the existence of solutions for 0 < lambda <= lambda(1), where lambda(1)>0 is the first eigenvalue of -Delta on Omega. Moreover, we prove the multiplicity of solutions for lambda >lambda(1) and b>0 in suitable intervals.