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 L²$L^2$-norm for a class of nonlinear Kirchhoff type problems in R³${\mathbb{R}}^3$ where a,b>0$a,b>0$ are constants, λ∈R$\lambda \in \mathbb{R}$, ge" 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">$p\in (\frac{14}{3},6)$. To get such solutions we look for critical points of the energy functional restricted on the following set For the value ge" 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">$p\in (\frac{14}{3},6)$ considered, the functional I_b$I_b$ is unbounded from below on S_r(c)$S_r\left(c\right)$. By using a minimax procedure, we prove that for any c>0$c>0$, there are infinitely many critical points ge" 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">${\left\{u_n^b\right\}}_{n\in {\mathbb{N}}^{+}}$ of I_b$I_b$ restricted on S_r(c)$S_r\left(c\right)$ with the energy ge" 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">$I_b\left(u_n^b\right)\to +\infty (n\to +\infty )$. Moreover, we regard b$b$ as a parameter and give a convergence property of ge" 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">$u_n^b$ as b→0⁺$b\to 0^{+}$.