Normalized Solutions of Nonautonomous Kirchhoff Equations: Sub- and Super-critical Cases

In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation -a+b∫R3|∇u|2dxΔu-λu=K(x)f(u),x∈R3;u∈H1(R3),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2{\mathrm {d}}x\right) \Delta u-\lambda u=K(x)f(u), &{} x\in {\mathbb {R}}^3; \\ u\in H^1({\mathbb {R}}^3), \end{array} \right. \end{aligned}$$\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}, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is unknown and appears as a Lagrange multiplier, K∈C(R3,R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in {\mathcal {C}}({\mathbb {R}}^3, {\mathbb {R}}^+)$$\end{document} with 0<lim|y|→∞K(y)≤infR3K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\lim _{|y|\rightarrow \infty }K(y)\le \inf _{{\mathbb {R}}^3} K$$\end{document}, and f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})$$\end{document} satisfies general L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-supercritical or L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm solutions of Kirchhoff problems.