Abstract

In this paper, we study the existence of solutions to the Kirchhoff equation \begin{equation*} - \Big ( {a + b\int_{{\mathbb{R}}^{3}} {|\nabla u{|^2}dx} } \Big ) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~{\rm in}~{\mathbb{R}}^{3}, \end{equation*} having prescribed mass $$ \int_{{\mathbb{R}}^{3}} {|u{|^2}dx} = c, $$ where $a,b > 0$, $\mu \in {\mathbb{R}}$, $2 < q < p < 6$. When $(p,q)$ belongs to a certain domain in ${{\mathbb{R}}^{2}}$, we prove the existence and nonexistence of normalized solutions by using constraint minimization and concentration compactness principle, our main results may be illustrated by the red areas and green areas shown in Figure 1. In particular, our results are closely related to the values of $\mu$ and prescribed mass $c$, and partially extend the results of Li et al. [10] and Soave [20].

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.