The existence of ground state normalized solution for mass supercritical modified Kirchhoff equation

Abstract

In this paper, we focus on the existence of ground state solution with prescribed $L^2$-norm for the following modified Kirchhoff problem: $-\left(a+b{\int }_{R^N}{|\nabla u|}^{2}dx\right)\Delta u-u\Delta \left(u^2\right)-\lambda u={\left|u\right|}^{p-2}u,x\in R^N,$where $N=2,3$, $a,b>0$ are constants, $p\in \left(4+\frac{4}{N},2^{\ast }\right)$, $2^{\ast }=6$ if $N=3$, and $2^{\ast }=+\infty$ if $N=2$. By employing a novel scaling method, we establish the existence of ground state normalized solutions for the above problem. Our result is new for the mass supercritical case $4+\frac{4}{N}<p\le 2^{\ast }$, notably for the case $p=2^{\ast }$.