Normalized Solutions to Nonautonomous Kirchhoff Equation

Abstract

In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with a perturbation: $$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda u=|u|^{p-2} u+h(x)\left |u\right |^{q-2}u, \quad \text{ in } \mathbb{R}^{N}, \\ &\int_{\mathbb{R}^{N}}\left|u\right|^{2}dx=c, \quad u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right. $$ where $1\le N\le 3, a,b,c>0, 1\leq q<2$, $\lambda \in \mathbb{R}$. We treat three cases. (i)When $2<p<2+\frac{4}{N},h(x)\ge0$, we obtain the existence of global constraint minimizers. (ii)When $2+\frac{8}{N}<p<2^{*},h(x)\ge0$, we prove the existence of mountain pass solution. (iii)When $2+\frac{8}{N}<p<2^{*},h(x)\leq0$, we establish the existence of bound state solutions.