On the concentration phenomenon of \(L^{2}\)-subcritical constrained minimizers for a class of Kirchhoff equations with potentials
type: Item
instance of: scholarly article
title: On the concentration phenomenon of \(L^{2}\)-subcritical constrained minimizers for a class of Kirchhoff equations with potentials
zbMATH Open document ID: 1423.35106
author: Gongbao Li
author: Hongyu Ye
published in: Journal of Differential Equations
publication date: 2019-03-22T00:00:00Z
review text: The authors study the existence and concentration phenomenon of normalized solutions to the following Kirchhoff equation \[ -\Bigl(a + b \int_{\mathbb{R}^N} |\nabla u|^2 \Bigr) \Delta u +(V (x)- \varrho)u - |u|^{p-2}u = 0, \quad x\in \mathbb{R}^N,\; \varrho\in \mathbb{R}, \tag{1.1} \] where \(N \le 3\), \(a, b > 0\) are constants and \(2 <p <2^* := \frac{2N}{N-2}\) and \(2^* =+\infty\) if \(N = 1, 2\). The potential \(V : \mathbb{R}^N \to \mathbb{R}\) is a suitable function. A solution \(u\) to (1.1) with \(|u|_2 = c\) corresponds to a critical point of the following \(C^1\)-functional \[ I_V (u) = \frac{1}{2} \int_{\mathbb{R}^N} (a|\nabla u|^2 + V(x)|u|^2)+ \frac{b}{4} \Bigl(\int_{\mathbb{R}^N} |\nabla u|^2 \Bigr)^2 - \frac{1}{p} \int_{\mathbb{R}^N} |u|^p,\quad u\in \mathcal{H}, \tag{1.2} \] on \[S_c = \Big\{u\in H^1(\mathbb{R}^N)| \int_{\mathbb{R}^N} V (x)|u|^2 < +\infty, |u|^2 =c >0\Big\}.\] The authors set \[ i_V (c) := \inf_{u\in S_c} I_V (u) \tag{1.3} \] and investigate the existence and concentration behavior of minimizers for \(i_V (c)\) when \(V(x)\) satisfies the following condition: \[ V(x) \in L^\infty_{loc}(\mathbb{R}^N), \quad \inf_{x\in\mathbb{R}^N} V (x) = 0 \quad\text{and } \lim_{|x|\to +\infty} V (x)=+\infty. \tag{\(V\)} \] The authors call \((u_c, \varrho_c) \in S_c \times \mathbb{R}\) a couple of solution to the equation (1.1) if \((u_c, \varrho_c)\) satisfies (1.1) and \(|u_c|_2 = c\). The first result is as follows. Theorem 1.2. Suppose that \(V(x)\) satisfies \((V)\), \(2 < p < 2^* \) and \(N \le 3\). \begin{itemize} \item[(1)] For each \[ \begin{cases} c > 0, & 2 < p < \frac{2N+8}{N} ,\\ 0 < c \le (\frac{b}{2}|Q_{\frac{2N+8}{N}}|^{\frac{8}{N}}_2 )^\frac{N}{8-2N}, & p= \frac{2N+8}{N} , \end{cases} \] \(i_V (c)\) has at least one minimizer. Moreover, there exists a couple of solution \((u_c, \varrho_c) \in S_c \times \mathbb{R}\) to (1.1) with \(I_V (u_c) = i_V (c)\). \item[(2)] For each \(c > (\frac{b}{2}|Q_{\frac{2N+8}{N}}|^{\frac{8}{N}}_2 )^…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 7040851
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: existence of normalized solutions
zbMATH Keywords: concentration
MaRDI profile type: Publication
full work available at URL: j.jde.2018.11.024
OpenAlex ID: W2964311100
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Wikidata QID: Q128901478
DOI: 10.1016/J.JDE.2018.11.024
description: scientific article; zbMATH DE number 7040851