TY - JOUR
T1 - Normalized Solutions for a Kirchhoff Equation with Potential in $\mathbb{R}^3$
AU - Xu , Yuan
AU - Lan , Yongyi
JO - Journal of Mathematical Study
VL - 4
SP - 509
EP - 527
PY - 2024
DA - 2024/12
SN - 57
DO - http://doi.org/10.4208/jms.v57n4.24.08
UR - https://global-sci.org/intro/article_detail/jms/23715.html
KW - Kirchhoff equation, normalized solutions, minimax principle.
AB - <p style="text-align: justify;">In this paper, for given mass $c&gt;0,$ we study the existence of normalized
solutions to the following nonlinear Kirchhoff equation&nbsp;&nbsp;$$\begin{cases} (a+b\int_{\mathbb{R}^3}[|\nabla u|^2+V(x)u^2]dx)[-\Delta u+V(x)u]=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u, \ \ \ {\rm in}\ \ \mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx=c^2, \end{cases}$$where $a&gt;0, b&gt;0, λ∈\mathbb{R},$ $5&lt;q&lt; p&lt;6,$ $\mu&gt;0$ and $V$ is a continuous non-positive function
vanishing at infinity. Under some mild assumptions on $V,$ we prove the existence of a
mountain pass normalized solution via the minimax principle.</p>