The existence of normalized solutions for L ²-critical constrained problems related to Kirchhoff equations

Abstract

In this paper, we study the existence of critical points for the following functional constrained on ${S_c=\{u\in H^1(\mathbb{R}^N)| |u|_2=c\}}$:

$$I(u)=\frac{a}{2}\int_{\mathbb{R}^N}|\nabla{u}|^{2}+\frac{b}{4}\left(\int_{\mathbb{R}^N}|\nabla{u}|^{2}\right)^{2}-\frac{N}{2N+8}\int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}},$$

where N = 1, 2, 3 and a, b > 0 are constants. The constraint problem is L ²-critical. We showed that I(u) has a constraint critical point with a mountain pass geometry on S _c if ${c > c^*:=(2^{-1}b|Q|_2^{\frac{8}{N}})^{\frac{N}{8-2N}}}$, where Q is the unique positive radial solution of ${-2\Delta Q+(\frac{4}{N}-1)Q=|Q|^{\frac{8}{N}} Q}$ in ${\mathbb{R}^N}$. For 0 < c < c ^*, I(u) has no critical point on S _c , and we proved the existence of minimizers for a new perturbation functional on S _c :

$$E_{a,b}(u)=\frac{a}{2} \int_{\mathbb{R}^N}|\nabla u|^2+\frac{b}{4} \left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^2-\frac{1}{4} \int_{\mathbb{R}^N}V(x)|u|^{4}-\frac{N}{2N+8} \int_{\mathbb{R}^N}|u|^{\frac{2N+8}{N}}.$$