Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials
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DOI：
10.1007/s00526-020-01831-4
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发表时间：
2019-09
影响因子：
2.1
通讯作者：
Lu Chen;Guozhen Lu;Maochun Zhu
中科院分区：
文献类型：
--
作者：
Lu Chen;Guozhen Lu;Maochun Zhu
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{2}\left( {\mathbb {R}}^{4}\right) $$\end{document} which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^{2}u+\gamma u=f(u)\ \text {in}\ {\mathbb {R}}^{4} \end{aligned}$$\end{document}and the range of, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( s\right) $$\end{document} is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^{2}u+V(x)u=\lambda s\exp (2|s|^{2}))\ \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$\end{document}whenV(x) is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the Eq. whenV(x) is the Rabinowitz type trapping potential, namely it satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0<\inf _{x \in {\mathbb {R}}^{4}} V(x)<\sup _{x \in {\mathbb {R}}^{4}} V(x)=\lim _{|x| \rightarrow +\infty } V(x). \end{aligned}$$\end{document}(See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in. (See Theorem 2.9.)
In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{2}\left( {\mathbb {R}}^{4}\right) $$\end{document} which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^{2}u+\gamma u=f(u)\ \text {in}\ {\mathbb {R}}^{4} \end{aligned}$$\end{document}and the range of, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( s\right) $$\end{document} is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^{2}u+V(x)u=\lambda s\exp (2|s|^{2}))\ \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$\end{document}whenV(x) is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the Eq. whenV(x) is the Rabinowitz type trapping potential, namely it satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0<\inf _{x \in {\mathbb {R}}^{4}} V(x)<\sup _{x \in {\mathbb {R}}^{4}} V(x)=\lim _{|x| \rightarrow +\infty } V(x). \end{aligned}$$\end{document}(See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in. (See Theorem 2.9.)