Our main results especially solve problem (0.1) in the case where mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si6.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=1d99826a0f087acccd280082b69e4197" title="Click to view the MathML source">p∈(2,3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem
where mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si8.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=13679829384bf6e78527d9bbc8fd9085" title="Click to view the MathML source">蔚>0 is a parameter, mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si9.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=cf0ce6e6a1fc18801abc47ecf6718a15" title="Click to view the MathML source">V(x) is a positive continuous potential and mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si10.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=c22335157a62039186e44fd5bb65c0ef" title="Click to view the MathML source">f(u)∼|u|p−1u with mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si11.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=1a08bb19cf70eae6e465c966f8dcf3bf" title="Click to view the MathML source">3<p<5 and satisfies the Ambrosetti–Rabinowitz type condition. Our main results extend also the arguments used in [7] and [33], which deal with Schrödinger–Poisson system with pure power nonlinearities, to the Kirchhoff type problem.