In this paper, we study the multiplicity of solutions with a prescribed
5=1caf88f83c5 880c73b6bc703f905 4989" title="Click to view the MathML source">L2 L 2 -norm for a class of nonlinear Kirchhoff type problems in
5=f42225 0ef231f81c944b726d55 320ed5 " title="Click to view the MathML source">R3 R 3 where
5" class="mathmlsrc">5.gif&_user=111111111&_pii=S1468121816300426&_rdoc=1&_issn=14681218&md5 =db7ac37e9904d328e45 401ae0c07e2e9" title="Click to view the MathML source">a,b>0 5.gif" overflow="scroll">a , b > 0 are constants,
5=2c7ad0de709cc6768e320c5 2d429d6cc" title="Click to view the MathML source">λ∈R λ ∈ R ,
5=df 7c982323f2d2dc5 466f94a6d05 eb3b"> 5" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si7.gif"> 5" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si7.gif"> p ∈ ( 14 3 , 6 ) . To get such solutions we look for critical points of the energy functional
restricted on the following set
For the value
5=df 7c982323f2d2dc5 466f94a6d05 eb3b"> 5" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si7.gif"> 5" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si7.gif"> p ∈ ( 14 3 , 6 ) considered, the functional
5=6803f172632cd0bcfcab95 982c9ca5 16" title="Click to view the MathML source">Ib I b is unbounded from below on
5=ccb37ca46a6497cf409725 027032c333" title="Click to view the MathML source">Sr (c) S r ( c ) . By using a minimax procedure, we prove that for any
5=35 48ef44eb8a13c8b834e15 e82f42b5 e" title="Click to view the MathML source">c>0 c > 0 , there are infinitely many critical points
5=aeddda1f499107f8ff5 e837cb62a1196"> 55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si14.gif"> 55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si14.gif"> { u n b } n ∈ N + of
5=6803f172632cd0bcfcab95 982c9ca5 16" title="Click to view the MathML source">Ib I b restricted on
5=ccb37ca46a6497cf409725 027032c333" title="Click to view the MathML source">Sr (c) S r ( c ) with the energy
5=7555 999291cb723ef0d16f1aa199f86b"> 51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si17.gif"> 51" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si17.gif"> I b ( u n b ) → + ∞ ( n → + ∞ ) . Moreover, we regard
5=f90c00699f75 35 1cdd435 1ecae1c9008" title="Click to view the MathML source">b b as a parameter and give a convergence property of
5=fd4c1dafbc076796d38071d975 38b492"> 5" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S1468121816300426-si19.gif"> 5" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1468121816300426-si19.gif"> u n b as
5=db06fb7b029d449b15 bed5 cbdc05 2e06" title="Click to view the MathML source">b→0+ b → 0 + .