Integrable dynamics and geometric conservation laws of hyperelastic strips

1.   Introduction and main results

1.1.   Background and motivation

In this paper, we mainly focus our interest on the existence and concentration of normalized solutions of the following nonlinear elliptic problem involving a Kirchhoff term:

where $a, \, b, \alpha$ are positive real numbers and $\sigma\in(0, 2)$, $\lambda$ is unkown and appears as a Lagrange multiplier. Equation (1.1) is related to the stationary solutions of

Equation (1.2) was first proposed by Kirchhoff in ^[13] and regarded as an extension of the classical D'Alembert's wave equation, which describes free vibrations of elastic strings. Kirchhoff-type problems also appear in other fields like biological systems. To better understand the physical background, we refer the readers to ^[1,2,4,14]. From a mathematical point of view, problem (1.1) is not a pointwise identity because of the appearance of the term $( {\int}_{\mathbb{R} ^{3} }^{} \left | \nabla u \right |^{2}) \Delta u$. Due to such a characteristic, Kirchhoff- type equations constitute nonlocal problems. Compared with the semilinear states (i.e., setting $b = 0$ in the above two equations), the nonlocal term creates some additional mathematical difficulties which make the study of such problems particularly interesting.

In the literature about the following related unconstrained Kirchhoff problems, there have been a lot of results on the existence and concentration of solutions for small values of $\varepsilon$.

In physics, such solutions are called the semiclassical states for small values of $\varepsilon$. In ^[10], the existence, multiplicity and concentration behavior of positive solutions to the Kirchhoff problem (1.3) have been studied by He and Zou, where $V(x)$ is a continuous function and $f$ is a subcritical nonlinear term. For the critical case, Wang et al., in ^[28] obtained some multiplicity and concentration results of positive solutions for the Kirchhoff problem (1.3). And He et al., in ^[11] obtained the concentration of solutions in the critical case. Recently, multi-peak solutions were established by Luo et al., in ^[18] for the following problem:

In ^[15] Li et al., revisited the singular perturbation problem (1.4), where $V(x)$ satisfies some suitable assumptions. They established the uniqueness and nondegeneracy of positive solutions to the following limiting Kirchhoff problem:

By the Lyapunov-Schmidt reduction method and a local Pohozaev identity, single-peak solutions were obtained for (1.4). In the past decades, other related results have also been widely studied, such as the existence of ground states, positive solutions, multiple solutions and sign-changing solutions to (1.4). We refer the reader to ^{[7,9,10,16,29]} and the references therein.

In recent years, the problems on normalized solutions have attracted much attention from many researchers. In ^[25,26], Stuart considered the problem given by

in the mass-subcritical case and obtained the existence of normalized solutions by seeking a global minimizer of the energy functional. In ^[12], Jeanjean considered the mass supercritical case and studied the existence of normalized solutions to problem (1.5) by using the mountain pass lemma. For the Sobolev critical case, Soave in ^[24] considered normalized ground state solutions of problem (1.5) with $f(u) = \mu|u|^{q-2}u+|u|^{2^\ast-2}u$, where $2^\ast = 2N/(N-2), N\ge3$ is the Sobolev critical exponent. For $f(u) = g(u)+|u|^{2^\ast-2}u$ with a mass critical or supercritical state but Sobolev subcritical nonlinearity $g$, we refer the reader to ^[19]. Now, we would like to mention some related results on Kirchhoff problems. The authors of ^[29,30] considered the problem in the mass subcritical and mass critical cases:

with $a, \, b > 0$ and $p\in(2, \, 2^{*})$. The existence and non-existence of normalized solutions are obtained. In ^[20], the Kirchhoff problem (1.6) was investigated for $f(u) = \mu|u|^{q-2}u+|u|^{2^\ast-2}u$ and $N = 3$. With the aid of a subcritical approximation approach, the existence of normalized ground states can be obtained for $\mu > 0$ large enough. Moreover, the asymptotic behavior of ground state solutions is also considered as $c\rightarrow \infty$. As for further results on Sobolev critical Kirchhoff equations and high energy normalized solutions, we refer the reader to ^[21,22,32].

In what follows, we turn our attention to normalized multi-bump solutions of the Kirchhoff problem (1.1). For the related results on Schrödinger equations, we refer the reader to the references ^[27,31]. In ^[31], the following nonlinear Schrödinger equation was studied by Zhang and Zhang:

For the case that the parameter $\hbar$ goes to $0$, the authors of ^[31] constructed normalized multi-bump solutions around the local maximum points of $K$ by employing the variational gluing methods of Séré ^[23] and Zelati and Rabinowitz ^[5,6], as well as the penalization technique ^[31]. Soon afterward, Tang et al., in ^[27] considered normalized solutions to the nonlinear Schrödinger problem

with an $L^{2}$-constraint. By taking the limit as $\lambda\to+\infty$, they derive the existence of normalized multi-bump solutions with each bump concentrated around the local minimum set of $a(x)$.

1.2.   Main result of this paper

Motivated by ^[27,31], the present paper is devoted to the existence and concentration behavior of the multi-bump solutions for the Kirchhoff problem (1.1). In contrast to the nonlinear Schrödinger problems, the Kirchhoff term brings us some additional difficulties. We intend to obtain the existence of multi-bump solutions for (1.1).

Before stating our main result, we give the following assumptions:

$(A)$ $\alpha \in (3, \, \frac{2}{\sigma })$ if $\sigma \in (0, \, \frac{2}{3})$ and $\alpha \in (\frac{2}{\sigma }, \, 3)$ if $\sigma \in (\frac{2}{3}, \, 2)$.

$(K)$ $K\in(\mathbb{R}^{3}, \, (0, \, +\infty))\cap L^{\infty}(\mathbb{R}^{3})$ and there are $\ell\ge 2$ mutually disjoint bounded domains $\Omega _{i}\subset\mathbb{R}^{3}$, $i = 1, \, 2, \, \cdots, \, \ell$ such that

Denote $\mathcal{K}_{i} = \left \{ x\in \Omega _{i}|K(x) = k_{i}\right \}$, which is nonempty and compact and set

Now, we state our main result as follows.

Theorem 1.1. Assume that $(A)$ and $(K)$. There is $\varepsilon _{0} > 0$ such that for each $\varepsilon\in (0, \, \varepsilon _{0})$, it follows that (1.1) admits a solution $(\lambda _{\varepsilon }, \, v_{\varepsilon})\in \mathbb{R}\times H^{1}(\mathbb{R}^{3})$ with the following properties:

$(a)$ $v_{\varepsilon}$ admits exactly $\ell$ local maximum points $P_{i, \, \varepsilon}$, $i = 1, \, 2, \, \cdots, \, \ell$ that satisfy

$(b)$ $\mu = \varepsilon^{\frac{2\sigma(3-\alpha)}{2-3\sigma}}\lambda_\varepsilon \to \mu_{0}$ and $\|\varepsilon^\frac{3-\alpha}{2-3\sigma}v_{\varepsilon}(\varepsilon^{\beta}\cdot)-\sum_{i = 1}^{\ell}u_{i}(\cdot-\varepsilon^{-\beta}P_{i, \, \varepsilon})\|_{H^{1}}\to0$ as $\varepsilon\to0$, where

and $U\in H^{1}(\mathbb{R}^{3})$ is a positive solution to

$(c)$ There are constants $C, \, c > 0$ that are independent of $\varepsilon$ such that

1.3.   The strategy for the proof

The proof of Theorem 1.1 is similar to that in ^[31]. By virtue of the change of variables techinque, we have

Equation(1.1) is transformed into the following problem:

Let

Then, under the assumption $(A)$ and given $\beta > 0$ and $d > 0$, we have the following:

Define the energy functional

Then, a solution $(\mu_{\hbar}, \, u_{\hbar})$ of (1.10) can be obtained as a critical point of $E_{\hbar}$ that is restrained on

By adopting similar deformation arguments in ^[5,6,23,31], we show that the Lagrange multiplier $\mu _{\hbar}$ satisfies

where $q_{i, \, \hbar}$ satisfies the condition that $\mbox{dist}(\hbar q_{i, \, \hbar}, \mathcal{K}_{i})\to0$ as $\hbar\to0$, $i = 1, \, 2, \, \cdots, \, \ell$.

This paper is organized as follows: In Section 2, we study the existence and variational structure of solutions to the limit equation of Eq (1.1). In Section 3, we introduce the penalized function which satisfies the Palais-Smale condition. In Section 4, we prove the existence of a critical point of the penalized function in the subcritical and supercritical cases. In Section 5, we show that the critical point is a solution to the original problem through the application of a decay estimate.

Notation : In this paper, we make use of the following notations:

● $|u|_p:=\left(\int_{\mathbb{R}^3}|u|^p\right)^{\frac{1}{p}}$, where $u\in L^{p}(\mathbb{R}^{3})$, $p\in[1, \, \infty)$;

● $\|u\|:=\left(\int_{\mathbb{R}^3}|\nabla u|^2+|u|^2\right)^{\frac{1}{2}}$, where $u\in H^{1}(\mathbb{R}^{3})$;

● $b^{\pm } = \max\left \{ 0, \, \pm b \right \}$ for $b\in \mathbb{R}$;

● $B(x, \, \rho)$ denotes an open ball centered at $x\in \mathbb{R}^{3}$ with radius $\rho > 0$;

● For a domain $D\subset \mathbb{R}^{3}$, we denote $\frac{1}{\hbar } D: = \left \{ x\in \mathbb{R}^{3}|\hbar x\in D \right \}$;

● Unless stated otherwise, $\delta$ and C are general constants.

2.   The limit system

Let $m_{0}, \, \theta_{1}, \, \theta _{2}, \, \cdots, \, \theta_{\ell}$ be a series of positive numbers. We consider the following system:

Next, we refer the reader to ^[31] to show Lemmas 2.1–2.3 as follows.

Lemma 2.1. For $\sigma \in \left (0, \, \frac{2}{3} \right)\cup \left (\frac{2}{3}, \, 2\right)$, system (2.1) has a unique solution $(\mu, \, v_{1}, \, v_{2}, \, \cdots, \, v_{\ell})\in\mathbb{R}\times H^{1}(\mathbb{R}^{3})^{\ell}$ up to translations of each $v_{i}, \, i = 1, \, 2, \, \cdots, \, \ell$, where

and $U\in H^{1}(\mathbb{R}^{3})$ is the unique positive radial solution to (1.9).

By using (2.2), we can obtain the mass distribution for each $v_{i}, \, i = 1, \, 2, \, \cdots, \, \ell$ in the limit system (2.1), as follows:

and for each $i = 1, \, 2, \, \cdots, \, \ell, \, v_{i}$ is the ground state of

on

Lemma 2.2. $\sum_{i = 1}^{\ell}I_{\theta _{i}}(v_{i})$ is continuous and strictly decreasing with respect to $m_{0}$ and $\theta_{i}, \, i = 1, \, 2, \, \cdots, \, \ell$, where $v_{i}$ is determined as in Lemma 2.1.

We next characterize the energy level of $\sum_{i = 1}^{\ell}I_{\theta_{i}}(v_{i})$. Let

and for each $s_{i} > 0,$ the minimizing problem

is achieved for each $i = 1, \, 2, \, \cdots, \, \ell$ given some radial function $w_{s_{i}}.$ In particular, $v_{i} = w_{s_{i}^{0}}$ for $s_{i}^{0} = \left|v_{i}\right|_{2}$. Moreover, if $\sigma\in\left(0, \, \frac{2}{3}\right)$, then

and if $\sigma\in\left(\frac{2}{3}, \, 2\right)$, then

Set

and define $E(s): = \sum_{i = 1}^{\ell}I_{\theta_{i}}(w_{s_{i}})$ for $s\in S_{+}^{\ell-1}.$

Lemma 2.3. Denote $s^{0} = (s_{1}^{0}, \, s_{2}^{0}, \, \cdots, \, s_{\ell}^{0}) = (\left|v_{1}\right|_{2}, \, \left|v_{2}\right|_{2}, \, \cdots, \, \left|v_{\ell}\right|_{2})$. For each $s\in S_{+}^{\ell-1}\setminus\left\{s^{0}\right\}$, the following statements hold:

$(a)$ If $\sigma \in (0, \, \frac{2}{3})$, then $\sum_{i = 1}^{\ell}I_{\theta_{i}}(v_{i}) = E(s^{0}) > E(s)$;

$(b)$ If $\sigma \in (\frac{2}{3}, \, 2)$, then $\sum_{i = 1}^{\ell}I_{\theta_{i}}(v_{i}) = E(s^{0}) < E(s)$.

3.   Existence of constrained localized Palais-Smale sequences

In this section, we adopt the penalization argument and the deformation approach in ^[31] to obtain a constrained localized Palais-Smale sequence. Denote $(\mu_{0}, u_{i})$ as the solution of the limit system (2.1) with $m_{0} = 1$ and $\theta_{i} = k_{i}, \, i = 1, \, 2, \, \cdots, \, \ell$, where $(k_{i})_{i = 1}^{\ell}$ denotes positive numbers given by $(K)$. Next, we set $b_{0}: = \sum_{i = 1}^{\ell}I_{i}(u_{i})$, where

Then, we will find a positive solution $(\mu_{\hbar}, u_{\hbar})$ to the following system:

satisfying

with $\hbar q_{i, \, \hbar}\to q_{i}\in\mathcal{K}_{i}.$

Set $\mathcal{M}: = \left \{ u\in H^{1}(\mathbb{R}^{3})|\left | u \right |_{2} = 1 \right \}$ and for $i = 1, \, 2, \, \cdots, \, \ell$ and $\tau > 0$, define

Define the following equation for each $\rho \in (0, \, \frac{1}{10}\min_{1\le i\le \ell}\left \| u_{i} \right \|_{L^{2} (B_{1}(0)) })$:

For $u\in H^{1}(\mathbb{R}^{3})$, consider the penalized energy functional $I_{\hbar }:H^{1}(\mathbb{R}^{3}) \to \mathbb{R}$ is given by

where

and

We also denote

Note that if $u_{\hbar}\in\mathcal{M}$ with $\left \| u_{\hbar} \right \|^{2}_{H^{1}(\mathbb{R}^{3}\setminus\cup _{i = 1}^{\ell}\frac{1}{\hbar }\Omega _{i})} < \hbar$ is a critical point of $I_{\hbar }|_{\mathcal{M} }$, then it solves (3.1) for some $\mu _{\hbar }$. Denote the tangent space of $\mathcal{M}$ at $u\in\mathcal{M}$ by

Lemma 3.1. For any $L\in\mathbb{R}$, there exists $\hbar_{L} > 0$ such that for any fixed $\hbar\in(0, \, \hbar_{L})$, if a sequence $\left \{ u_{n, \, \hbar } \right \}\subset Z(\rho)$ such that

as $n\to\infty$, then $u_{n, \, \hbar }$ has a strong convergent subsequence in $H^{1}(\mathbb{R}^{3})$.

Proof. Set $u_{n, \, \hbar } = { \sum_{i = 1}^{\ell}}u_{i}(x-z_{n, \, i, \, \hbar})+v_{n, \, \hbar }$ with $\hbar z_{n, \, i, \, \hbar}\in(\mathcal{K}_{i})^{\tau }$ and $\left \| v_{n, \, \hbar } \right \| \le \rho$. It follows from $u_{n, \, \hbar}\in Z(\rho)$ that $\left \| u_{n, \, \hbar } \right \| \le \rho+ { \sum_{i = 1}^{\ell}}\left \| u_{i} \right \|$, which is bounded. Then, by

we have that $G_{\hbar }(u_{n, \, \hbar })\le I_{\hbar }(u_{n, \, \hbar })+\frac{1}{2\sigma +2} {\int}_{\mathbb{R}^{3} }^{}\mathrm {} K(\hbar x)\left | u_{n, \, \hbar } \right |^{2\sigma +2}\le C_{L}$ for some $C_{L} > 0$ that is independent of $\hbar$ and $n$. From the assumption (3.2), for some $\mu _{n, \, \hbar }\in\mathbb{R}$, we deduce that

We have

where $C_{L}^{*} > 0$ is independent of $\hbar$ and $n.$ Then up to a subsequence, $\mu_{n, \, \hbar }\to \mu _{\hbar }$ in $\mathbb{R}$ and $u_{n, \, \hbar }\rightharpoonup u_{\hbar} = { \sum_{i = 1}^{\ell} }u_{i}(x-z_{i, \, \hbar }) +v_{\hbar }$ in $H^{1}(\mathbb{R}^{3})$ with $z_{n, \, i, \, \hbar}\to z_{i, \, \hbar}\in\frac{1}{\hbar}(\mathcal{K}_{i})^{\tau}$ and $v_{n, \, \hbar}\rightharpoonup v_{\hbar}$.

Next, for any $\varphi \in H^{1}(\mathbb{R}^{3})$, note that $\underset{n \to \infty}\lim I'_{\hbar}(u_{n, \, \hbar})\varphi+\mu_{n, \, \hbar}J'(u_{n, \, \hbar})\varphi = 0$, $(\mu _{\hbar }, u _{\hbar })$ satisfies

where $Q_{\hbar } = 4\hbar ^{-1} \lim_{n \to \infty} G_{\hbar }(u_{n, \, \hbar })^{\frac{1}{2} }\ge 0$. Then, we claim that $\hbar_{L}$ and $\mu_{L}$ are two positive constants such that $\mu_{\hbar} > \mu_{L}$ for each $\hbar\in\left (0, \, \hbar_L \right)$. Otherwise, we assume that $\mu_{\hbar}\to\mu\le0$ as $\hbar\to0$ up to a subsequence. Because $u_{\hbar}$ is bounded in $H^{1}(\mathbb{R}^{3})$, we can assume that $u_{\hbar }(\cdot+z_{1, \, \hbar})\rightharpoonup u$. Note that

We can obtain that $u\ne 0$ if $\rho > 0$ is small. Then set $\varphi = \psi (x-z_{1, \, \hbar})$ in (3.4) for each $\psi \in C_{0}^{\infty}(\mathbb{R}^{3})$ and take the limit $\hbar\to0$, that is

Using the boundedness of ${u_{\hbar}}$ and $d > 0$, we have

We see that $u$ is a nontrivial solution to $-a\Delta u+\mu u = k_{0}\left | u \right |^{2\sigma}u$ in $H^{1}(\mathbb{R}^{3})$ for some $k_{0} > 0$, which is impossible by Lemma 2.1.

Setting $\varphi = u_{n, \, \hbar}-u_{\hbar}$ in (3.4), we have

Then it follows from (3.3) that

That is,

We can show that for $n$ large enough,

where using the fact that $u_{n, \, \hbar}\rightharpoonup u_{\hbar}$ in $H^{1}(\mathbb{R}^{3})$, it follows ${\int}_{\mathbb{R }^{3} }^{}\nabla u_{\hbar } \nabla (u_{n, \, \hbar}-u_{\hbar})\to0$. Thus from (3.5)–(3.7), we have

Noting also that ${\int}_{\mathbb{R}^{3}}K(\hbar x)\left| u_{n, \, \hbar}-u_{\hbar}\right|^{2\sigma+2}\le C||u_{n, \, \hbar}-u_{\hbar}||^{2\sigma+2}$ and

the following inequality holds:

where $C^{*}$ is a positive constant since $a > 0$ and $\mu_{\hbar} > 0$. Making $\rho$ smaller if necessary given $C||u_{n, \, \hbar}-u_{\hbar}||^{2\sigma} < C^{*}/2$, it follows that $u_{n, \, \hbar}\to u_{\hbar}$ in $H^{1}(\mathbb{R}^{3})$. This completes the proof of Lemma 3.1. □

Proposition 3.2. For some $\rho > 0$ small and by letting $\left \{ \hbar _{n}\right \}\subset\mathbb{R}$, $\left \{ \mu_{n}\right \}\subset\mathbb{R}$ and $\left \{u _{n}\right \}\subset Z(\rho)$ satisfy that

as $n\to\infty$. Then, $\mu _{n}\to\mu _{0}$ holds, $\lim_{n \to \infty } I_{\hbar _{n}} (u_{n}) = b_{0}$ and for some $z_{n, \, i}\in \mathbb{R}^{3},$ $i = 1, \, 2, \, \cdots, \, \ell,$ we have

Proof. The proof is similar to that in ^[31]. For the sake of completeness, we shall give the details.

Step 1. We claim that $\mu _{n}\to\overset{\sim}{\mu} > 0$.

As $\left \{u _{n}\right \}\subset Z(\rho)$, we can write that $u_{n} = { \sum_{i = 1}^{\ell}}u_{i}(x-z_{n, \, i })+v_{n}$ with $z_{n, \, i}\in\frac{1}{\hbar}(\mathcal{K}_{i})^{\tau }$ and $\left \| v_{n} \right \| \le \rho$. It follows from $u_{n}\in Z(\rho)$ and the boundedness of $I_{\hbar _{n} } (u_{n})$ that $\left \| u_{n} \right \|$ and $G_{\hbar_{n} }(u_{n})$ are bounded. Besides, by (3.9) and $J'(u_{n})u_{n} = 1$, we know that $\mu _{n}$ is bounded. Then up to a subsequence, we can assume that $\mu _{n}\to \overset{\sim}{\mu}$ in $\mathbb{R}$ and $u_{n}(\cdot +z_{n, \, i})\rightharpoonup w_{i}\in H^{1}(\mathbb{R}^{3})$. For $\rho < \frac{1}{10}\min_{1\le i\le \ell}\left \| u_{i} \right \|_{L^{2}(B_{1}(0)) }$, we have

Notice that for any $R > 0$, we can obtain that $\left \| u_{i}-w_{i} \right\|_{L^{2}(B_{R}(0))}\le\rho$. Hence,

Then, if $\rho$ is small enough, we know that $w_{i} \ne0$. Next, testing (3.9) with $\varphi(x-z_{n, \, i})$ for each $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$, we deduce that

Thus, $w_{i}$ is a solution to $-a\Delta w_{i}+\overset{\sim}{\mu} w_{i} = \overset{\sim}{k_{i}}\left | w_{i} \right |^{2\sigma}w_{i}$ in $H^{1}(\mathbb{R}^{3})$ with $\underset{n\to\infty}\lim K(\hbar_{n}z_{n, i}) \to\overset{\sim}{k_{i}} \in[\underline{k}, \, \bar{k}]$, where $\underline{k} = \min _{x \in U_{i = 1}^{\ell} \bar{\Omega}_{i}} K(x) > 0$ and $\bar{k} = \max _{1 \leq i \leq \ell} k_{i}$. Then, combining the Pohozaev identity with

it follows that there exists a positive contant $\overset{\sim }{\mu}$.

Step 2. $u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i}) \to 0\ \mbox{in}\ L^{2\sigma+2}(\mathbb{R}^{3})\ \mbox{and}\ \mbox{dist}(\hbar _{n}z_{n, \, i}, \, \mathcal{K}_{i})\to 0$.

We show that

Otherwise, by Lions' lemma ^[17], there exists a sequence of points $\left \{ z_{n} \right \}\subset\mathbb{R}^{3}$ such that

Noting that $\left | z_{n}- z_{n, \, i} \right |\to \infty$ $i = 1, \, 2, \, \cdots, \, \ell$, we have

By (3.8), $G_{\hbar_{n}}(u_{n})\le C$ holds for some $C > 0$ that is independent of $\hbar$. Then, we have that $\mbox{ dist}\left (\hbar_{n}z_{n}, \, \cup _{i = 1}^{\ell}\Omega_{i} \right)\to0$. Up to a subsequence, we assume that $\tilde{v}_{n}(x+z_{n})\rightharpoonup v_{0}\ne0$ in $H^{1}(\mathbb{R}^{3})$ and $K(\hbar_{n}z_{n}) \to k_{0}\in[\underline{k}, \, \bar{k}]$, where $k_{0} = k(y_{0}), \, y_{0}\in\cup _{i = 1}^{\ell}\Omega_{i}$. Let $D: = \left\{x\in\mathbb{R}^{3}|x_{3}\ge-M\right\}$. For some $i_{0}$, if $\underset{n \to \infty}\lim\frac{\mbox{dist}(\hbar_{n} z_{n}, \, \partial\Omega_{{i}_{0}})}{\hbar_{n}} = M < \infty$, we get that $\hbar_{n}z_{n}\to z_{0}$ as $n\to\infty$, where $z_{0}\in\partial\Omega_{{i}_{0}}$. Next, without loss of generality we can assume that $v_{0}\in H_{0}^{1}(D)$. Testing (3.9) with $\varphi (\cdot-z_{n})$ for any $\varphi\in C_{0}^{\infty}(D)$, we have

Then by applying $\left\|u_{n}\right\|_{{H}^{1}(\mathbb{R}^{3}\setminus\frac{1}{\hbar_{n}}\cup_{i = 1}^{\ell}\Omega_{i})}\le C\hbar_{n}$ and

we can obtain that $v_{0}$ is a solution of $-a\Delta u +\overset{\sim}{\mu} u = k_{0}\left | u\right |^{2\sigma}u$ in $H_{0}^{1}(D)$, which is impossible since this equation does not have a nontrivial solution on the half space according to ^[8]. Thus $\underset{n \to \infty} \lim\mbox{ dist}(\hbar_{n} z_{n}, \, \partial\Omega_{{i}_{0}}) = +\infty$ and $z_{n}\in\frac{1}{\hbar_{n}}\Omega_{{i}_{0}}$. Now we test (3.9) with $\varphi (\cdot-z_{n})$ for any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ to get

where $\overset{\sim}{\mu} > 0$, and $\left | v_{0} \right |_{2}^{2} > C_{1}$ for some $C_{1} > 0$ that is independent of $\rho$.

If we have chosen $\rho$ small enough, then by the Brézis-Lieb lemma,

which is a contradiction.

Step 3. $\left \| u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i}) \right \|\to 0\ \mbox{and}\ \lim_{n \to \infty } I_{\hbar_{n}} (u_{n}) = b_{0}$.

Testing (3.9) with $u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i})$, given

we can get that

Next, we have

i.e., $u_{n}-\sum_{i = 1}^{\ell}w_{i}(\cdot-z_{n, \, i})\to 0\ \mbox{in}\ H^{1}(\mathbb{R}^{3})$.

On the other hand, by Lemma 2.2, we obviously get that $\underset{n \to \infty } \lim I_{\hbar_{n}} (u_{n}) = b_{0}$. □

4.   Existence of critical points

In this section, let $\rho$ be fixed in Proposition 3.2. We present the result as follows.

Proposition 4.1. There exists $\hbar _{0} > 0$ such that for $\hbar\in(0, \, \hbar _{0})$, $I_{\hbar }|_{\mathcal{M} }$ has a critical point $u_{\hbar }\in Z(\rho)$. Moreover, $\lim_{\hbar \to 0}I(u _{\hbar }) = b_{0}$ and the Lagrange multiplier $\mu _{\hbar }\in\mathbb{R}$ satisfies

Remark 4.2. By Proposition 3.2, it is easy to verify that (4.1) holds if $u_{\hbar}$ is a critical point of $I_{\hbar}|_\mathcal{M}$ such that $\limsup _{\hbar\to0}I_{u_{\hbar}}\le b_{0}$.

The proof of Proposition 4.1 can be obtained as in ^[31] by considering the following contradiction: $\left \{ \hbar_{n} \right \}$ with $\hbar_{n}\to0$ such that for some sequence ${b_{\hbar_{n}}}\to b_{0}$, $I_{\hbar}$ admits no critical points in $\left \{ u\in Z(\rho)|I_{\hbar_{n}}(u)\le b_{\hbar_{n}} \right \}$. For brevity, we denote $\hbar = \hbar_{n}.$ Then from Lemma 3.1 and Proposition 3.2, there respectively exist $\kappa _{0} > 0$ and $v > 0$ independent of $\hbar$ and $v_{\hbar} > 0$ such that

where

for any $b_{1}, \, b_{2}\in \mathbb{R}$.

Thanks to (4.2), one can get the following deformation lemma.

Lemma 4.3. Let $v_{\hbar}$ and $v$ be given as in (4.2). For any $\kappa\in\left(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\}\right),$ there exists $\hbar_{\kappa} > 0$ such that for $\hbar\in \left(0, \, \hbar_{\kappa}\right)$ there is a deformation $\eta:\mathcal{M}\to\mathcal{M}$ that satisfied the following conditions:

$(a)$ $\eta(u) = u$ if $u\in\mathcal{M}\setminus (Z(\rho)\cap \left [ b_{0}-2\kappa \le I_{\hbar} \right ]).$

$(b)$ $\ I_{\hbar}(\eta (u))\le I_{\hbar}(u)$ if $u\in\mathcal{M}.$

$(c)$ $\eta(u)\in Z(\rho)\cap \left [ I_{\hbar}\le b_{0}-\kappa \right ]$ if $u\in Z(\rho/4)\cap\left [ I_{\hbar}\le b_{\hbar} \right ].$

To give the proof of Lemma 4.3, we borrow some ideas from ^[5,6,31] in the $L^{2}$-subcritical case and $L^{2}$-supercritical case.

4.1.   $L^{2}$-subcritical case $\sigma \in \left (0, \frac{2}{3} \right)$

For every $\delta > 0$, we denote

where $s^{0} = (|u_{1}|_{2}, \, \cdots, \, |u_{\ell}|_{2})$. Fix $q_{i}\in \mathcal{K}_{i}$ and $q_{i, \, \hbar} = \frac{1}{\hbar}q_{i}$ for $i = 1, \, 2, \, \cdots, \, \ell$ and define the $(\ell-1)$-dimensional initial path by

where $B_{\hbar}: = \left | \sum_{i = 1}^{\ell}w_{s_{i}}(\cdot-q_{{i}, \, \hbar}) \right |_{2}^{-1}$. Note that we can fix $\delta > 0$ small enough such that

and

Define

Lemma 4.4. $\lim_{\hbar\to0}b_{\hbar} = b_{0}$ and fix any $\kappa\in(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\})$ such that

where $\partial S_{\delta}: = \left\{s\in S_{+}^{\ell}|\left|s-s^{0}\right| = \delta\right\}$.

Proof. Since

one can deduce that

By Lemma 2.3(a), we have

□

Proof of Proposition 4.1 in the $L^{2}$-subcritical case. By Lemma 4.3 and (4.3), we have

Define

Similar to the case in ^[31], there exists $s^{1}\in S_{\delta}$ such that $\Psi_{i, \, \hbar}(\eta(\xi_{\hbar}(s^{1}))) = s_{i}^{0} = \left|u_{i}\right|_{2}$. Denote

where $\gamma _{i, \, \hbar}\in C_{0}^{\infty}(\frac{1}{\hbar}(\Omega_{i}^{'}), \, \left[0, \, 1\right])$ is a cut-off function such that $\gamma _{i, \, \hbar} = 1$ on $\frac{1}{\hbar}\Omega_{i}$ and $\left|\nabla\gamma_{i, \, \hbar}\right|\le C\hbar$ for each $i = 1, \, 2, \, \cdots, \, \ell$ and some $C > 0$; $\Omega_{i}^{'}$ is an open neighborhood of $\bar{\Omega}_{i}$. By (4.5), we have that $G_{\hbar}(u_{0, \, \hbar})\le C$ for some $C > 0$ that is independent of $\hbar$, which implies that

Then

and

Hence from (4.5)–(4.9), we have

which is a contradiction. This completes the proof. □

4.2.   $L^{2}$-supercritical case $\sigma \in \left (\frac{2}{3}, 2\right)$

Fix $q_{i}\in \mathcal{K}_{i}$ and denote $q_{i, \, \hbar} = \frac{1}{\hbar}q_{i}$; we set

where $\bar{B}_{\hbar}: = \left |{ \sum_{i = 1}^{\ell}}{t_{i}^{3/2} }u_{i}(t_{i} (\cdot-q_{i, \, \hbar})) \right |_{2}^{-1}$.

Define

Note that we can fix $\bar{\delta } > 0$ small enough such that

and $\bar{B}_{\hbar}\to1$ holds. Note also that

Since

and

one can get the result as in ^[31].

Lemma 4.5. $\lim_{\hbar\to0}b_{\hbar} = b_{0}$ and fix any $\kappa\in(0, \, \min\left\{\kappa_{0}, \, \frac{\rho v}{16}\right\})$ such that

Proof of Proposition 4.1 in the $L^{2}$-supercritical case. By Lemma 4.3 and (4.10),

Define

Similar to the case in ^[31], there exists $t^{1}\in[1-\bar{\delta}, \, 1+\bar{\delta}]^{\ell}$ such that

We denote

Similar to (4.7) and (4.8), we have

and

From (4.13)–(4.15), we have

A direct calculation shows that

and

Hence by the definition of $b_{s_{i}}$, we have

Similarly, one can get a contradiction. □

5.   Completion of the proof

Let $u_{\hbar}$ be the critical point of the modified function $I_{\hbar}$ given in Proposition 4.1.

Completion of proof of Theorem 1.1.

Proof. We show that there exists $c > 0$ independent of $\hbar$ such that

We adopt some arguments from ^[3,31]. Set $\left \lfloor 2\hbar ^{-1}\tau \right \rfloor-1: = n_{\hbar }$. For $n = 1, \, 2, \, \cdots, \, n_{\hbar }$, we take $\phi _{n}\in C^{1}(\mathbb{R }^{3}, \, \left [ 0, \, 1 \right ])$ such that

where $E_{n}: = \left \{ x\in \mathbb{R}^{3}|\mbox{dist}(x, \, \cup _{i = 1}^{\ell}\frac{1}{ \hbar}(\mathcal{K}_{i})^{\frac{\tau}{2} }) > n-1\right \}$. Then by Proposition 3.2,

Note that for each $n = 1, \, 2, \, \cdots, \, n_{\hbar }$,

Since $\left \langle I'_{\hbar }(u_{\hbar })+\mu_{\hbar }J'(u_{\hbar }), \, \phi _{n}u_{\hbar }\right \rangle = 0$, we have

Therefore, by (5.3) and the Sobolev embedding,

where $-\hbar^{d}b {\int}_{\mathbb{R} ^{3} }^{ }\left | \nabla u_{\hbar } \right |^{2} {\int}_{\mathbb{R} ^{3} }\nabla u_{\hbar }\nabla(\phi _{n} u_{\hbar })\le o_{\hbar}(1)(2||u_{\hbar}||^{2}_{H^{1}(E_{n})}-||u_{\hbar}||^{2}_{H^{1}(E_{n+1})})$ as $\hbar\rightarrow0$. By (5.2), we have

where

Nothing that $n_{\hbar }\ge\frac{ \tau }{\hbar}$ for small values of $\hbar$, one can take some $\theta_0 > 1$ and obtain

It follows that for small values of $\hbar$, $G_{\hbar}(u_{\hbar}) = 0$. So $u_{\hbar}$ is a solution to the original problem (3.1) for small values of $\hbar$. □

Author contributions

Zhidan Shu: Writing-original draft and Writing-review & editing; Jianjun Zhang: Methodology and Supervision. All authors equally contributed to this manuscript and approved the final version.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referee for his/her valuable suggestions and comments.

Conflict of interest

The authors declare no conflicts of interest.