Multiplicity and asymptotic behavior of normalized solutions to $ p $-Kirchhoff equations

In this paper, we explored the following $ p $-Kirchhoff equation with the $ L^p $-mass constraint

$ \begin{equation} \nonumber \begin{cases} -\left( a+b\int_{\mathbb{R} ^3}{\left| \nabla u \right|^pdx} \right) \Delta _pu = \lambda \left| u \right|^{p-2}u+\left| u \right|^{q-2}u, \ x \in \mathbb{R}^3 \\ \left(\int_{\mathbb{R} ^3}{\left| u \right|^{p}dx}\right)^\frac{1}{p} = c > 0, \end{cases} \end{equation} $

where $ a > 0 $, $ b > 0 $, $ \frac{3}{2} < p < 3 $, $ p < q < p^{\ast}: = \frac{3p}{3-p} $, $ \Delta _pu = div\left(\left| \nabla u \right|^{p-2}\nabla u \right) $, and $ \lambda \in \mathbb{R} $ appears as a Lagrange multiplier. We considered $ L^p $-mass subcritical, $ L^p $-mass critical, and $ L^p $-mass supercritical cases. Making full use of the Gagliardo-Nirenberg inequality, we established the existence of the minimizers in $ L^p $-mass subcritical and $ L^p $-mass critical cases and the nonexistence in the $ L^p $-mass critical case. For the $ L^p $-mass supercritical case, a radially symmetric ground state was obtained using the Nehari-Pohozaev manifold method. On the other hand, employing a $ \sigma $-homotopy stable argument of fountain theorem type, we obtained infinitely many radial normalized solutions. In addition, the asymptotic behavior of normalized solutions as $ b \rightarrow 0^+ $ was also explored. Finally, we proved the uniqueness of normalized solutions and provided precise descriptions when $ \frac{3}{2} < p \leq 2 $.