Normalized solutions for Kirchhoff type equations with combined nonlinearities: The Sobolev critical case.

In this paper, we study the Kirchhoff equation with Sobolev critical exponent$ -\left(a+b\int_{ {\mathbb{R}}^3}|\nabla u|^2\right)\Delta u = \lambda u+\mu|u|^{q-2}u+|u|^{4}u\ \ {\rm in}\ {\mathbb{R}}^3 $under the normalized constraint$ \int_{ {\mathbb{R}}^3}u^2 = c^2, $where $ a, \, b, \, c>0 $ are constants, $ \lambda, \, \mu\in{\mathbb{R}} $ and $ 2<q<6 $. The number $ 2+8/3 $ behaves as the $ L^2 $-critical exponent for the above problem. When $ \mu>0 $, we distinguish the problem into four cases: $ 2<q<2+4/3 $, $ q = 2+4/3 $, $ 2+4/3<q<2+8/3 $ and $ 2+8/3\leq q<6 $, and prove the existence and multiplicity of normalized solutions under suitable assumptions on $ \mu $ and $ c $. The solution obtained is either a minimum (local or global) or a mountain pass solution. When $ \mu\leq 0 $, we establish the nonexistence of nonnegative normalized solutions. Finally, we investigate the asymptotic behavior of normalized solutions obtained above as $ \mu\to0^+ $ and as $ b\to0^+ $ respectively. [ABSTRACT FROM AUTHOR]