On the Kirchhoff equation with prescribed mass and general nonlinearities
Xiaoyu Zeng, Jianjun Zhang, Yimin Zhang, Xuexiu Zhong
Abstract
In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $ (\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N) $ to the following Kirchhoff problem$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+\lambda u = g(u)\; \hbox{in}\; \mathbb{R}^N, \;N\geq 1 $satisfying the normalization constraint $ \int_{\mathbb{R}^N}u^2 = c, $ which appears in free vibrations of elastic strings. The parameters $ a, b>0 $ are prescribed as well as the mass $ c>0 $. Due to the presence of the non-local term $ b\int_{\mathbb{R}^N}|\nabla u|^2dx \Delta u $, such problems lack the mountain pass geometry in the higher dimension case $ N\geq 5 $. Our result seems to be the first attempt in this aspect.