Normalized solutions to nonautonomous Kirchhoff equation

Abstract

In this paper, we studied 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 treated three cases:(i) When $ 2 < p < 2+\frac{4}{N}, h(x)\ge0 $, we obtained the existence of a global constraint minimizer.(ii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\ge0 $, we proved the existence of a mountain pass solution.(iii) When $ 2+\frac{8}{N} < p < 2^{*}, h(x)\leq0 $, we established the existence of a bound state solution.

Keywords

• nonautonomous kirchhoff equations
• normalized solutions
• bound state solution
• $ l^{2} $-critical exponent