Taiwanese Journal of Mathematics

Remarks on Normalized Solutions for $L^2$-Critical Kirchhoff Problems

Yonglong Zeng and Kuisheng Chen

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Abstract

We study a constraint minimization problem on $S_c = \{ u \in H^1(\mathbb{R}^N), |u|_2^2 = c, c \in (0, c^*) \}$ for the following $L^2$-critical Kirchhoff type functional:\begin{align*}  E_\alpha(u)  &= \frac{a}{2} \int_{\mathbb{R}^N} |\nabla u|^2 \, dx    + \frac{b}{4} \left( \int_{\mathbb{R}^N} |\nabla u|^2 \, dx \right)^2    + \frac{1}{\alpha+2} \int_{\mathbb{R}^N} V(x) |u|^{\alpha+2} \, dx \\  &\quad - \frac{N}{2N+8} \int_{\mathbb{R}^2} |u|^{\frac{2N+8}{N}} \, dx,\end{align*}where $N \leq 3$, $a, b \gt 0$ are constants, $\alpha \in [0, \frac{8}{N})$ and $V(x) \in L^\infty(\mathbb{R}^N)$ is a suitable potential. We prove that the problem has at least one minimizer if $\alpha \in [2, \frac{8}{N})$ and the energy of the minimization problem is negative. Moreover, some non-existence results are obtained when the energy of the problem equals to zero.

Article information

Source
Taiwanese J. Math., Volume 20, Number 3 (2016), 617-627.

Dates
First available in Project Euclid: 1 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1498874470

Digital Object Identifier
doi:10.11650/tjm.20.2016.6548

Mathematical Reviews number (MathSciNet)
MR3511999

Zentralblatt MATH identifier
1357.35135

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Keywords
Kirchhoff equation $L^2$-critical minimization problems variational method

Citation

Zeng, Yonglong; Chen, Kuisheng. Remarks on Normalized Solutions for $L^2$-Critical Kirchhoff Problems. Taiwanese J. Math. 20 (2016), no. 3, 617--627. doi:10.11650/tjm.20.2016.6548. https://projecteuclid.org/euclid.twjm/1498874470


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