Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

被引:0
作者
Shenghao Feng
Jianhua Chen
Jijiang Sun
Xianjiu Huang
机构
[1] Nanchang University,Department of Mathematics
来源
Fractional Calculus and Applied Analysis | 2024年 / 27卷
关键词
Kirchhoff–Schrödinger–Poisson; Critical problem; Steep potential well; 35A15; 35J20; 35J61;
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摘要
In this paper, we consider the following Kirchhoff-Schrödinger-Poisson equation: a+b[u]s2(-Δ)su+Vλ(x)u+ϕu=|u|p-2u+|u|2s∗-2uinR3,(-Δ)tϕ=u2inR3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{lc} \left( a+b[u]_s^2\right) (-\varDelta )^s u+V_\lambda (x) u+\phi u=|u|^{p-2}u+|u|^{2_s^*-2} u &{}{} \text { in } {\mathbb {R}}^3, \\ (-\varDelta )^t \phi =u^2 &{}{} \text { in } {\mathbb {R}}^3, \end{array}\right. \end{aligned} \end{aligned}$$\end{document}where s∈34,1,t∈(0,1),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( \frac{3}{4}, 1\right) , t \in (0,1),$$\end{document}p>4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>4,$$\end{document}Vλ(x)=λV(x)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_\lambda (x)=\lambda V(x)+1$$\end{document} with λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} and [u]s2=∫R3∫R3|u(x)-u(y)|2|x-y|3+2sdxdy.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}{}[u]_s^2=\int _{{\mathbb {R}}^3} \int _{{\mathbb {R}}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2 s}} \text {d} x \text {d} y. \end{aligned} \end{aligned}$$\end{document}Under some conditions on V, when λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} large enough and b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>0$$\end{document} small enough, we use the deformation lemma and constrained minimization arguments to prove the existence of least energy sign-changing solutions. Additionally, we prove the least energy sign-changing solutions is strictly larger than twice that the ground state energy. In particular, a further analysis of the phenomenon of concentration for least energy sign-changing solutions as λ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow \infty $$\end{document} and b→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \rightarrow 0$$\end{document}.
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页码:124 / 156
页数:32
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