In this paper, we study
the multiplicity of solutions with a prescribed
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si2.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=1caf88f83c5880c73b6bc703f9054989" title ="Click to view the MathML source">L2 class="mathContainer hidden">class="mathCode">L 2 -norm for a
class of nonlinear Kirchhoff type problems in
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si1.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=f422250ef231f81c944b726d55320ed5" title ="Click to view the MathML source">R3 class="mathContainer hidden">class="mathCode">R 3 class="formula" id="fd000005">
class="mathml">
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_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−2 u, class="mathContainer hidden">class="mathCode">− ( a + b ∫ R 3 | ∇ u | 2 ) Δ u − λ u = | u | p − 2 u , class="temp" src="/sd/blank.gif">
where
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si5.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=db7ac37e9904d328e45401ae0c07e2e9" title ="Click to view the MathML source">a,b>0 class="mathContainer hidden">class="mathCode">a , b > 0 are constants,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si6.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=2c7ad0de709cc6768e320c52d429d6cc" title ="Click to view the MathML source">λ∈R class="mathContainer hidden">class="mathCode">λ ∈ R ,
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si7.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=df7c982323f2d2dc5466f94a6d05eb3b"> class="imgLazyJSB inlineImage" height="18" width="65" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si7.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si7.gif"> class="mathContainer hidden">class="mathCode">p ∈ ( 14 3 , 6 ) . To get such solutions we look for critical points of
the energy functional
class="formula" id="fd000010">
restricted on
the following set
class="formula" id="fd000015">
For
the value
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si7.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=df7c982323f2d2dc5466f94a6d05eb3b"> class="imgLazyJSB inlineImage" height="18" width="65" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si7.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si7.gif"> class="mathContainer hidden">class="mathCode">p ∈ ( 14 3 , 6 ) considered,
the functional
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si11.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=6803f172632cd0bcfcab95982c9ca516" title ="Click to view the MathML source">Ib class="mathContainer hidden">class="mathCode">I b is unbounded from below on
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si12.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=ccb37ca46a6497cf409725027032c333" title ="Click to view the MathML source">Sr (c) class="mathContainer hidden">class="mathCode">S r ( c ) . By using a minimax procedure, we prove that for any
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si13.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=3548ef44eb8a13c8b834e15e82f42b5e" title ="Click to view the MathML source">c>0 class="mathContainer hidden">class="mathCode">c > 0 ,
the re are infinitely many critical points
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si14.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=aeddda1f499107f8ff5e837cb62a1196"> class="imgLazyJSB inlineImage" height="20" width="55" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si14.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si14.gif"> class="mathContainer hidden">class="mathCode">{ u n b } n ∈ N + of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si11.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=6803f172632cd0bcfcab95982c9ca516" title ="Click to view the MathML source">Ib class="mathContainer hidden">class="mathCode">I b restricted on
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si12.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=ccb37ca46a6497cf409725027032c333" title ="Click to view the MathML source">Sr (c) class="mathContainer hidden">class="mathCode">S r ( c ) with
the energy
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si17.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=7555999291cb723ef0d16f1aa199f86b"> class="imgLazyJSB inlineImage" height="17" width="151" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si17.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si17.gif"> class="mathContainer hidden">class="mathCode">I b ( u n b ) → + ∞ ( n → + ∞ ) . Moreover, we regard
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si18.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=f90c00699f75351cdd4351ecae1c9008" title ="Click to view the MathML source">b class="mathContainer hidden">class="mathCode">b as a parameter and give a convergence property of
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si19.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=fd4c1dafbc076796d38071d97538b492"> class="imgLazyJSB inlineImage" height="17" width="15" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si19.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si19.gif"> class="mathContainer hidden">class="mathCode">u n b as
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1468121816300426&_mathId=si20.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5=db06fb7b029d449b15bed5cbdc052e06" title ="Click to view the MathML source">b→0+ class="mathContainer hidden">class="mathCode">b → 0 + .