Existence and asymptotic behavior of normalized solutions for the modified Kirchhoff equations in $ \mathbb{R}^3 $

Abstract

This paper is concerned with the following modified Kirchhoff type problem $ \begin{align*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u-u\Delta (u^2)-\lambda u=|u|^{p-2}u, \; \; \; x\in \mathbb{R}^3, \end{align*} $ where $ a, b > 0 $ are constants and $ \lambda\in \mathbb R $. When $ p=\frac{16}{3} $, we prove that the existence of normalized solution with a prescribed $ L^2 $-norm for the above equation by applying constrained minimization method. Moreover, when $ p\in\left(\frac{10}{3}, \frac{16}{3}\right) $, we prove the existence of mountain pass type normalized solution for the above modified Kirchhoff equation by using the perturbation method. And the asymptotic behavior of normalized solution as $ b\rightarrow 0 $ is analyzed. These conclusions extend some known ones in previous papers.

Keywords

• modified kirchhoff-type equations
• l²-normalized solutions
• asymptotic behavior
• constrained minimization method
• perturbation method