Existence and concentration behavior of normalized solutions for critical Kirchhoff type equations with general nonlinearities

We consider the following Kirchhoff equation in the Sobolev critical case with combined power nonlinearities

having prescribed mass

$$\begin{aligned} \mathop {\int }\limits _{{\mathbb {R}}^{3}}|u|^2 =c^2, \end{aligned}$$

where \(a,\ c,\ \mu >0\) are positive constants, \(b>0\) is a positive parameter, \(2<q<{\bar{p}}:=2+\frac{8}{3}\) which is \(L^{2}\)-critical exponent. For the \(L^{2}\)-subcritical case \(2<q<\frac{10}{3}\) and Sobolev critical case, Li et al. (2021) proved that \(({\mathcal {K}})\) has a solution which is ground state solution and corresponds to local minima of the associated energy functional. Here we extend the result in Li et al. (2021) by proving that \(({\mathcal {K}})\) has the second solution which is not a ground state and is located at a mountain-pass level of the energy functional. Meanwhile, let \(u_{b}\) are normalized solutions of mountain-pass type to \(({\mathcal {K}})\), then \(u_{b}\rightarrow u\) in \(H^{1}({\mathbb {R}}^{3})\) as \(b\rightarrow 0\) up to a subsequence, where \(u\in H^{1}({\mathbb {R}}^{3})\) is a normalized solution of mountain-pass type to

$$\begin{aligned} -a\triangle u =\lambda u+ \mu |u|^{q-2}u +|u|^{4}u\ \ \ \ \ \ \ \textrm{in} \ {{\mathbb {R}}^{3}}. \end{aligned}$$

Our results also extend the results of Soave (J Differ Equ 269:6941–6987, 2020; J Funct Anal 279:108610, 2020).