In this paper, we study the multiplicity of solutions with a prescribed 426&_mathId=si2.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=1caf88f83c5880c73b6bc703f9054989" title="Click to view the MathML source">L2-norm for a class of nonlinear Kirchhoff type problems in 426&_mathId=si1.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=f422250ef231f81c944b726d55320ed5" title="Click to view the MathML source">R3
426&_mathId=si4.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=463457d47c4f4a0224f5d8cad2475321" title="Click to view the MathML source">−(a+b∫R3|∇u|2)Δu−λu=|u|p−2u,
where 426&_mathId=si5.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=db7ac37e9904d328e45401ae0c07e2e9" title="Click to view the MathML source">a,b>0 are constants, 426&_mathId=si6.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=2c7ad0de709cc6768e320c52d429d6cc" title="Click to view the MathML source">λ∈R, 426&_mathId=si7.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=df7c982323f2d2dc5466f94a6d05eb3b">426-si7.gif">. To get such solutions we look for critical points of the energy functional