Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $ L^2 $-subcritical nonlinearities

Abstract

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:$ \begin{equation*} \begin{cases} -\left(a\varepsilon^2+b\varepsilon\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u = \mu u+f(u) & {\rm{in}}\;\mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx = m\varepsilon^3 , u\in H^1(\mathbb{R}^3) , \end{cases} \end{equation*} $where $ a $, $ b $, $ m > 0 $, $ \varepsilon $ is a small positive parameter, $ V $ is a nonnegative continuous function, $ f $ is a continuous function with $ L^2 $-subcritical growth and $ \mu\in\mathbb{R} $ will arise as a Lagrange multiplier. Under the suitable assumptions on $ V $ and $ f $, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $ V $ attained its minimum value.

Keywords

• kirchhoff type equation
• normalized solutions
• multiplicity
• mass subcritical
• lusternik- schnirelman category