Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Super-Critical Case

In this paper, we study the existence of normalized solution to the following nonlinear mass super-critical Kirchhoff equation -a+b∫RN|∇u|2▵u+V(x)u+λu=g(u)inRN0≤u∈Hr1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u+V(x)u+\lambda u=g(u) \ \ {in} \ {{\mathbb {R}}^{N}}\\&0\le u\in H^{1}_{r}({\mathbb {R}}^{N}) \end{aligned}\right. \end{aligned}$$\end{document}where a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a ,b>0$$\end{document} are constants, λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in R$$\end{document}, and V(x) satisfies appropriate assumptions; g has a mass super-critical growth when N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document}, and g(u)=|u|p-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(u)=|u|^{p-2}u$$\end{document} with p∈(2+8N,2∗),2∗=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (2+\frac{8}{N},2^{*}), 2^{*}=\frac{2N}{N-2}$$\end{document} when N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}. Here, we prove the existence of ground state normalized solution via variational methods.