Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent

In this paper, we consider the existence and asymptotic properties of solutions to following Kirchhoff equation \(- \left(a+b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}}\right) \Delta u=\lambda u+ {| u |^{p - 2}}u+\mu |^{q 2}}u\) in \(\mathbb{R}^{3}\) under normalized constraint \(\int_{{\mathbb{R}^3}} {{u}^2}=c^2\), where \(a>0\), \(b>0\), \(c>0\), \(2<q<\frac{14}{3}<p\leq 6\) or \(\frac{14}{3}<q< p\leq 6\), \(\mu>0\) \(\lambda\in\mathbb{R}\) appears as a Lagrange multiplier. both cases for range \(p\) \(q\), Sobolev critical exponent \(p=6\) is involved corresponding energy functional unbounded from below on \(S_c=\{ \in H^{1}({\mathbb{R}^3})\colon \int_{{\mathbb{R}^3}} {{u}^2}=c^2 \}\). If \(2<q<\frac{10}{3}\) \(\frac{14}{3}<p<6\), obtain multiplicity result equation. \(2<q<\frac{10}{3}<p=6\) get ground state solution Furthermore, derive several results obtained solutions. Our extend Soave (J. Differential Equations 2020 & J. Funct. Anal. 2020), which studied nonlinear Schrödinger equations with combined nonlinearities, equations. To deal special difficulties created by nonlocal term \(({\int_{{\mathbb{R}^3}} {\left| \right|} ^2}) u\) appearing type equations, develop perturbed Pohozaev approach find way clear picture profile fiber map via careful analysis. meantime, need some subtle estimates \(L^2\)-constraint recover compactness case.