Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in

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• 作者：Xiao Luo^a ; ^{luoxiaohf@163.com" class="auth_mail" title="E-mail the corresponding author} ; Qingfang Wang^b ; ^{hbwangqingfang@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Kirchhoff type ; Normalized solutions ; Asymptotic behavior ; Variational methods
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：33
• 期：Complete
• 页码：19-32
• 全文大小：680 K

文摘

In this paper, we study the multiplicity of solutions with a prescribed mathmlsrc">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">L²mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">L2math>-norm for a class of nonlinear Kirchhoff type problems in mathmlsrc">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">R³mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">mathvariant="double-struck">R3math> where mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll">a,b>0math> are constants, mathmlsrc">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∈RmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">λ∈mathvariant="double-struck">Rmath>, mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll">p∈(143,6)math>. To get such solutions we look for critical points of the energy functional restricted on the following set For the value mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll">p∈(143,6)math> considered, the functional mathmlsrc">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">I_bmathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll">Ibmath> is unbounded from below on mathmlsrc">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">S_r(c)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll">Sr(c)math>. By using a minimax procedure, we prove that for any mathmlsrc">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>0mathContainer hidden">mathCode"><math altimg="si13.gif" overflow="scroll">c>0math>, there are infinitely many critical points mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si14.gif" overflow="scroll">{unb}n∈mathvariant="double-struck">N+math> of mathmlsrc">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">I_bmathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll">Ibmath> restricted on mathmlsrc">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">S_r(c)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll">Sr(c)math> with the energy mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si17.gif" overflow="scroll">Ib(unb)→+∞(n→+∞)math>. Moreover, we regard mathmlsrc">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">bmathContainer hidden">mathCode"><math altimg="si18.gif" overflow="scroll">bmath> as a parameter and give a convergence property of mathmlsrc">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">mage" 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">mathContainer hidden">mathCode"><math altimg="si19.gif" overflow="scroll">unbmath> as mathmlsrc">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⁺mathContainer hidden">mathCode"><math altimg="si20.gif" overflow="scroll">b→0+math>.