On the concentration phenomenon of \(L^{2}\)-subcritical constrained minimizers for a class of Kirchhoff equations with potentials
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 \(c > (\frac{b}{2}|Q_{\frac{2N+8}{N}}|^{\frac{8}{N}}_2 )^\frac{N}{8-2N}\) when \(p = \frac{2N+8}{N}\) or \(c > 0\) when \(\frac{2N+8}{N} < p < 2^*\), \(i_V (c) = -\infty\) and \(i_V (c)\) has no minimizer. \end{itemize} The main result of this paper is the following theorem. Theorem 1.4. Suppose that \(V(x)\) satisfies \((V)\), \(N\le 3\) and \(p \in (2, \frac{2N+8}{N} )\setminus\{4\}\). For any sequence \(\{c_n\}\subset (0, +\infty)\) with \(c_n\to \infty\) as \(n \to \infty\), let \((u_{c_n} , \varrho_{c_n} ) \in S_{c_n}\times \mathbb{R}\) be a couple of solution to (1.1) obtained in Theorem 1.2. Then there exists a subsequence of \(\{c_n\}\) (still denoted by \(\{c_n\}\)) and a sequence \(\{z_n\} \in \mathbb{R}^N\) such that \[ \frac{|Q_p|_2}{c_n}\Bigl(\frac{c_n}{m_{c_n}}\Bigr)^\frac{N}{2} u_{c_n}\Bigl(\frac{c_n}{m_{c_n}}(x +z_n)\Bigr) \to Q_p(x) \] in \(L^q (\mathbb{R}^N)\) for all \(2 \le q <2^*\) and \[ \Bigl[\frac{1}{c_n} \bigl(\frac{c_n}{m_{c_n}}\bigr)^\frac{N}{2}\Bigr]^{p-2} \varrho_{c_n}\to -\frac{4|Q_p|^{p-2}_2}{2N- p(N-2)} \] as \(n \to+\infty\).
scientific article; zbMATH DE number 7040851
scientific article