Normalized Solutions to Nonautonomous Kirchhoff Equation

Abstract

In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with a perturbation: $$\{\begin{array}{rl}& -\left(a+b{\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{2}dx\right)\Delta u+\lambda u=|u{|}^{p-2}u+h(x){\left|u\right|}^{q-2}u,\text{ in }{\mathbb{R}}^N,\\ & {\int }_{{\mathbb{R}}^N}{\left|u\right|}^{2}dx=c,u\in H^1({\mathbb{R}}^N),\end{array}$$ where $1\le N\le 3,a,b,c>0,1\le q<2$, $\lambda \in \mathbb{R}$. We treat three cases. (i)When $2<p<2+\frac{4}{N},h(x)\ge 0$, we obtain the existence of global constraint minimizers. (ii)When $2+\frac{8}{N}<p<2^{\ast },h(x)\ge 0$, we prove the existence of mountain pass solution. (iii)When $2+\frac{8}{N}<p<2^{\ast },h(x)\le 0$, we establish the existence of bound state solutions.

arXiv admin note: text overlap with arXiv:2301.07926 by other authors