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

详细信息    查看全文

• 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 pan id="mmlsi2" class="mathmlsrc">pan 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">L²pan>pan class="mathContainer hidden">pan class="mathCode">$L^2$pan>pan>pan>-norm for a class of nonlinear Kirchhoff type problems in pan id="mmlsi1" class="mathmlsrc">pan 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">R³pan>pan class="mathContainer hidden">pan class="mathCode">${\mathbb{R}}^3$pan>pan>pan> where pan id="mmlsi5" class="mathmlsrc">pan 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>0pan>pan class="mathContainer hidden">pan class="mathCode">$a,b>0$pan>pan>pan> are constants, pan id="mmlsi6" class="mathmlsrc">pan 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∈Rpan>pan class="mathContainer hidden">pan class="mathCode">$\lambda \in \mathbb{R}$pan>pan>pan>, pan id="mmlsi7" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$p\in (\frac{14}{3},6)$pan>pan>pan>. To get such solutions we look for critical points of the energy functional restricted on the following set For the value pan id="mmlsi7" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$p\in (\frac{14}{3},6)$pan>pan>pan> considered, the functional pan id="mmlsi11" class="mathmlsrc">pan 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">I_bpan>pan class="mathContainer hidden">pan class="mathCode">$I_b$pan>pan>pan> is unbounded from below on pan id="mmlsi12" class="mathmlsrc">pan 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">S_r(c)pan>pan class="mathContainer hidden">pan class="mathCode">$S_r\left(c\right)$pan>pan>pan>. By using a minimax procedure, we prove that for any pan id="mmlsi13" class="mathmlsrc">pan 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>0pan>pan class="mathContainer hidden">pan class="mathCode">$c>0$pan>pan>pan>, there are infinitely many critical points pan id="mmlsi14" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">${\left\{u_n^b\right\}}_{n\in {\mathbb{N}}^{+}}$pan>pan>pan> of pan id="mmlsi11" class="mathmlsrc">pan 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">I_bpan>pan class="mathContainer hidden">pan class="mathCode">$I_b$pan>pan>pan> restricted on pan id="mmlsi12" class="mathmlsrc">pan 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">S_r(c)pan>pan class="mathContainer hidden">pan class="mathCode">$S_r\left(c\right)$pan>pan>pan> with the energy pan id="mmlsi17" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$I_b\left(u_n^b\right)\to +\infty (n\to +\infty )$pan>pan>pan>. Moreover, we regard pan id="mmlsi18" class="mathmlsrc">pan 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">bpan>pan class="mathContainer hidden">pan class="mathCode">$b$pan>pan>pan> as a parameter and give a convergence property of pan id="mmlsi19" class="mathmlsrc">pan class="mathContainer hidden">pan class="mathCode">$u_n^b$pan>pan>pan> as pan id="mmlsi20" class="mathmlsrc">pan 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⁺pan>pan class="mathContainer hidden">pan class="mathCode">$b\to 0^{+}$pan>pan>pan>.