In this paper, we study the fo
llowing Kirchhoff type equation with critica
l growth
lass="formu
la" id="fd000005">
where
lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si2.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=33fb950e2aefa5dbfdb99496994fc062" title="Click to view the MathML source">a>0,b≥0lass="mathContainer hidden">lass="mathCode"> and
lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si3.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=398727f33b6213eb3973e75ca1685996" title="Click to view the MathML source">Ωlass="mathContainer hidden">lass="mathCode"> is a smooth bounded domain in
lsi4" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si4.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=ebd42ef49dd29d2195a05563abeb98a9" title="Click to view the MathML source">R3lass="mathContainer hidden">lass="mathCode">. When the rea
l parameter
lsi5" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si5.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=cc0e9ea79a02c1bc6e6fa900b1777f5f" title="Click to view the MathML source">μlass="mathContainer hidden">lass="mathCode"> is
larger than some positive constant, we investigate the mu
ltip
licity of nontrivia
l so
lutions for the above prob
lem with parameter
lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si6.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=f67cf5f117fda276f3542cb282631913" title="Click to view the MathML source">λlass="mathContainer hidden">lass="mathCode"> be
longing to a
left neighborhood of the Dirich
let eigenva
lue of the Lap
lacian operator
lsi7" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302233&_mathId=si7.gif&_user=111111111&_pii=S0893965916302233&_rdoc=1&_issn=08939659&md5=338b67f8abe230159956e416aba2ebef" title="Click to view the MathML source">−△lass="mathContainer hidden">lass="mathCode">.