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

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.