Multiplicity and concentration of normalized solutions for a Kirchhoff type problem with $L^2$-subcritical nonlinearities

1.   Introduction

In 1883, Kirchhoff ^[1] first proposed the following time-dependent wave equation:

as an extension of the classical D'Alembert's wave equations for the free vibration of elastic strings, where $u$ is the transverse displacement, $\rho$ is the mass density, $h$ is the cross-sectional area, $L$ is the length, $E$ is Young's modulus, and $P_0$ is the initial axial tension. The Kirchhoff equation (1.1) has attracted the attention of many researchers since Lions proposed an abstract functional analysis framework. Some interesting results can be referenced, for example, in ^[2,3,4].

In the past several years, there have been a lot of interesting results for the following Kirchhoff type problem:

Li and Ye ^[5] studied (1.2) with $f(u) = |u|^{p-1}u$, $2 < p < 5$ under some suitable assumptions on $V$. By employing a monotonicity trick and a new version of the global compactness lemma, they proved a positive ground state solution. In ^[6], combining the nondegeneracy result and Lyapunov-Schmidt reduction method, for $\varepsilon > 0$ sufficiently small, Li et al. ^[6] obtained the existence of solutions to problem (1.2) with $f(u) = |u|^{p-1}u$, $p\in(1, 5)$. If $V(x)$ and $\lambda f(u)+|u|^4u$ are replaced by $M(x)$ and $f(u)$, respectively, where $\lambda > 0$ is a parameter, $f$ is a continuous superlinear and subcritical nonlinearity. Using minimax theorems and the Ljusternik-Schnirelmann theory, Wang et al. ^[7] proved that for $\lambda > 0$ enough large and $\varepsilon > 0$ enough small, there exists a positive ground state solution, and they also verified the number of positive solutions in connection with the topology of the set of the global minima of the potentials.

He and Zou considered the Kirchhoff equation (1.2) in ^[8] with $f(u)$ satisfying the Ambrosetti-Rabinowitz condition and $V(x)$ satisfying

$(V_1)$ $\inf_{x\in\mathbb{R}^3}V(x) < \liminf_{|x|\to\infty}V(x)$.

Through Ljusternik-Schnirelmann theory and minimax methods, He and Zou ^[8] first obtained the abstract framework and some compactness properties of the functional associated to (1.2), then proved the number of solutions with the topology of the set where $V$ attains its minimum. Under general conditions of $f$, the potential function $V(x)$ is nonnegative and has $k$ sets of local minima in $\mathbb{R}^3$. By variational methods, Hu and Shuai ^[9] bear out the existence of multi-peak solutions to singularly prturbed Kirchhoff problems (1.2). Besides, readers can find some interesting results about (1.2) in ^{[10,11,12,13,14]} and the references therein.

In the past decade, normalized solutions, that is, solutions with the prescribed $L^2$ norm, to several problems have been received plenty of attention. From the point of view of physics, this approach seems to be more meaningful since it offers a better insight into the dynamical properties of the stationary solutions, for example, stability or instability. Recently, the study on the normalized solutions to the following Kirchhoff type equation involving an $L^2$ constraint:

has also been the purpose of very active research, where $a, b, m > 0$ are the given constants, $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier, and $p\in(2, 6)$. It is clear that solutions to (1.3) correspond to critical points of the functional $\Phi:H^1(\mathbb{R}^3)\to\mathbb{R}$ defined by

constrained to the sphere

Moreover, it is well-known that the study of (1.3) and the type of results one can expect depend on $p$. In particular, the range of $p$ determines whether the functional $I$ is bounded from below on $S_m$ and impacts on the choice of the approaches to search for constrained critical points. Roughly speaking, in the $L^2$-subcritical case, i.e., $p\in\left(2, \frac{14}{3}\right)$, one may use a minimization on $S_m$ in order to obtain the existence of a global minimizer; in the $L^2$-supercritical case, i.e., $p\in\left(\frac{14}{3}, 6\right)$, $I$ is unbounded from below on $S_m$ for any $m > 0$ and more efforts are always needed. We refer the reader to ^{[15,16,17,18,19,20,21,22,23,24,25,26]} and references therein.

A strong motivation to study multiplicity and concentration of normalized solutions to some Kirchhoff type equations mainly comes from the concentration phenomena for the following constrained singularly perturbed nonlinear Schrödinger equation:

Setting $u(x) = v(\varepsilon x)$, equation (1.4) will become the following equivalent equation:

In ^[27], Alves and Thin supposed $V$ satisfies the following conditions:

$(V)$ $V\in C(\mathbb{R}^N, \mathbb{R})\cap L^\infty(\mathbb{R}^N)$, $V(0) = 0$, and

and $f\in C^1(\mathbb{R}, \mathbb R)$ satisfying the following assumptions:

$(\tilde{f}_1)$ $f$ is odd and there are $q\in\left(2, 2+\frac{4}{N}\right)$ and $\alpha\in(0, +\infty)$ such that $\mathop{\lim}\limits_{s\to0}\frac{|f(s)|}{|s|^{q-1}} = \alpha > 0$.

$(\tilde{f}_2)$ There exist constants $c_1, c_2, c_3, c_4 > 0$, and $p\in\left(2, 2+\frac{4}{N}\right)$ such that

$(\tilde{f}_3)$ There is $q_1\in\left(2, 2+\frac{4}{N}\right)$ such that $\frac{f(s)}{s^{q_1-1}}$ is an increasing function of $s$ on $(0, +\infty)$.

Alves and Thin ^[27] demonstrated the existence of multiple normalized solutions to the class of elliptic problems (1.5) and the relation between the numbers of normalized solutions and the topology of the set where the potential $V$ attains its minimum value by minimization techniques and the Lusternik-Schnirelmann category.

After that, Alves and Thin ^[28] studied the class of elliptic problems (1.4) by assuming the conditions that $V\in C(\mathbb R^{3}, \mathbb{R})\cap L^\infty(\mathbb{R}^{3})$ satisfies

$(AT)$ $V(x)\ge0$ for all $x\in\mathbb{R}^{3}$ and there exists a bounded set $\Lambda\subset\mathbb{R}^{3}$ such that

and the nonlinearity $f$ is a continuous function with an $L^2$-subcritical growth and satisfies the following assumptions:

$(f'_1)$ $f$ is odd and $\lim_{t\to0} \frac{|f(s)|}{|s|^{q_0-1}} = \alpha > 0$ for some $q_0\in(2, 2+\frac{4}{N})$;

$(f'_2)$ there are constants $c_1, c_2 > 0$, and $p\in\left(2, 2+\frac{4}{N}\right)$ such that

$(f'_3)$ there is $q\in[q_0, 2+\frac{4}{N})$ such that $\frac{f(s)}{s^{q-1}}$ is an increasing function of $s$ on $(0, +\infty)$.

Through minimization techniques, the Lusternik-Schnirelmann category, and the penalization method, Alves and Thin ^[28] showed the existence of multiple normalized solutions to problem (1.4). Besides, they also proved the concentration of solutions. We mention that the geometry $(AT)$ considered in ^[27] does imply that potential $V$ has a global minimum. For other research results on (1.4), we refer readers to ^{[29,30,31,32,33,34,35]} and the references therein.

To the best of our knowledge, so far few results on the existence and multiplicity of normalized solutions are known to the singularly perturbed Kirchhoff problems involving an $L^2$ constraint. Inspired by Alves and Thin ^[27], we are interested in investigating the multiplicity and concentration of solutions to the following Kirchhoff type equation with $L^2$-constraint:

where $a, b, m, \varepsilon > 0$, $\mu$ is an unknown parameter that appears as a Lagrange multiplier, and $V\in C(\mathbb R^{3}, \mathbb{R})\cap L^\infty(\mathbb{R}^{3})$ satisfies $(V)$ and $V(x) = V(|x|)$. In the following, we will suppose that the nonlinearity $f$ satisfies the $L^2$-subcritical growth assumptions. More precisely, we introduce the following assumptions:

$(f_1)$ $f\in C(\mathbb{R}, \mathbb{R})$, $\lim_{s\to0} \frac{f(s)}{s} = 0$;

$(f_2)$ $\limsup_{|s|\rightarrow \infty}\frac{|f(s)|}{\left|s\right|^{11/3}} = 0$;

$(f_3)$ there exists $\zeta\neq0$ such that $F(\zeta) > 0$;

$(f_{4})$ $\liminf_{s\rightarrow0}\frac{F(s)}{\left|s\right|^{10/3}} = +\infty$;

$(f'_{4})$ $\limsup_{s\rightarrow0}\frac{F(s)}{\left|s\right|^{10/3}} < +\infty$.

We would like to point out that in ^[36], under the so-called Berestycki-Lions type mass subcritical growth conditions assumptions: $(f_3)$ and

$(HS_1)$ $f\in C(\mathbb{R}, \mathbb{R})$, $\lim_{s\to0} \frac{f(s)}{s} = 0$, and $\limsup_{|t|\rightarrow \infty}\frac{|f(t)|}{|t|^5} < \infty$;

$(HS_2)$ $\limsup_{t\rightarrow \infty}\frac{F(t)}{\left|t\right|^{14/3}}\leq0$,

which are weaker than $(f_1)$ and $(f_2)$, if $f$ satisfies $(f_{4})$ or $(f'_{4})$, for given mass $m > 0$, Hu and Sun studied the existence and nonexistence of constrained minimizers of the energy functional

on $S_m = \left\{u\in H^1(\mathbb{R}^3):\, \|u\|^2_2 = m\right\},$ where $a, b > 0$. They also established the relationship between the normalized ground state solutions and the ground state to the action functional $I(u)-\frac{\lambda}{2}\|u\|_2^2$.

To illustrate our results, we provide some notations. Define

and

where $\delta > 0$ and $dist(x, M)$ denotes the usual distance in $\mathbb{R}^3$ between $x$ and $M$.

Now we state our main result.

Theorem 1.1. Suppose that $f$ is odd and satisfies the conditions $(f_1)-(f_3)$ and $V\in C(\mathbb R^{3}, \mathbb{R})\cap L^\infty(\mathbb{R}^{3})$ satisfies $(V)$ and $V(x) = V(|x|)$. If $(f_{4})$ holds, we set $m > 0$. If $(f'_{4})$ holds, we assume $m > m^*$ for some $m^* > 0$. Then, for each $\delta > 0$ small enough, the following properties holds:

$(1)$ There exist $\varepsilon_0 > 0$ and $\Theta_m > 0$ such that for $0 < \varepsilon < \varepsilon_0$ and $\|V\|_\infty\leq\Theta_m$, (1.6) admits at least $cat_{M_\delta}(M)$ couples $(u_j, \mu_j)\in H_r^1(\mathbb{R}^3)\times\mathbb{R}$ of weak solutions with $\int_{\mathbb{R}^3}|u_j|^2dx = m\varepsilon^3$ and $\mu_j < 0$.

$(2)$ Let $u_\varepsilon$ denote one of these solutions and $\xi_\varepsilon$ is the global maximum of $|u_\varepsilon|$, then

Remark 1.1. (i) As in ^[37], if $Y$ is a closed subset of a topological space $X$, the Lusternik-Schnirelmann category $cat_{X}(Y)$ is the least number of closed and contractible sets in $X$ which cover $Y$. If $X = Y$, we use the notation $cat(X)$.

(ii) It is worth mentioning that our assumptions are much weaker than ^[27] and we do not need the monotonicity condition $(\widetilde{f}_3)$, which plays a crucial role in verifying the compactness of the Palais-Smale sequences. However, due to the existence of nonlocal term, the arguments to prove the compactness of certain bounded Palais-Smale sequences in ^[27] cannot be used directly even if the monotonicity condition holds. To overcome this difficulty, in the present paper, we work in the radial subspace of $H^1(\mathbb{R}^3)$ and more subtle analyses are required. It is an open question whether problem (1.6) admits a solution without the radially symmetric assumption on $V$. Moreover, it is interesting to study the existence and concentration phenomena of solutions under the local assumption $(AT)$.

(iii) Due to the existence of the nonlocal term, in contrast to the mass constrained nonlinear Schrödinger equations in ^[27], the behavior of $f$ near $0$ for the Kirchhoff type equation depends heavily on the growth rate $\frac{10}{3}$ and not the mass critical exponent $\frac{14}{3}$. Thus, in the present paper, we discuss the two different cases $(f_{4})$ and $(f'_{4})$ separately.

To begin with, in order to prove our Theorem 1.1, we set $u(x) = w(\varepsilon x)$, and equation (1.6) is equivalent to the following equation:

We also show the energy functional:

where $F(t) = \int_{0}^{t}f(s)ds$. In addition, we denote by $I_0, I_\infty:H_r^1(\mathbb{R}^3)\to\mathbb{R}$ the following functionals

and

It is clear that we need to prove at least that $cat_{M_\delta}(M)$ couples $(u_i, \mu_i)\in H_r^1(\mathbb{R}^3)\times\mathbb{R}$ solutions to (1.7) correspond to critical points of the energy functional $I$ constrained to the sphere

The paper is organized as follows. In Section 2, we study some technique results. In Section 3, we prove the energy functional $I_\varepsilon$ on the sphere $S_{m}$ satisfies the Palais-Smale condition at some negative level, and then prove Theorem 1.1 via the Lusternik-Schnirelmann category theory.

Notation. Throughout this paper, we denote by $c, c_i, C, C_i, C'_i, C', i = 1, 2, \cdots$ for various positive constants whose exact value may change from lines to lines but are not essential to the analysis of the problem. $\|u\|_{q} = \left(\int_{\mathbb{R}^3}|u|^qdx\right)^{\frac{1}{q}}$ denotes the usual norm of $L^{q}(\mathbb{R}^3)$ for $q\in[2, \infty)$, and $\|u\| = \left(\int_{\mathbb{R}^3}(|\nabla u|^2+|u|^2)dx\right)^{\frac{1}{2}}$ denotes the usual norm in the Sobolev space $H^1(\mathbb{R}^3)$. $H_r^1(\mathbb{R}^3)$ denotes the radial subspace of $H^1(\mathbb{R}^3)$. We use "$\rightarrow$" and "$\rightharpoonup$" to denote the strong and weak convergence in the related function space, respectively. We will write $o(1)$ to denote quantity that tends to $0$ as $n\to\infty$.

2.   Some technical results

In this section, inspired by ^[27,30], we will give some technical results which are useful to study our problem. First of all, we consider the existence of the normalized solution for the autonomous problem:

where $a, b, m > 0, \mathcal{V}\ge0$, and $\mu\in\mathbb{R}$ is represented as a Lagrange multiplier, $f$ is a continuous function satisfying $(f_1)-(f_3)$, and either $(f_4)$ or $(f'_4)$ holds. As is known to all, solutions to (2.1) correspond to critical points of the functional $I_{\mathcal{V}}:H_r^1(\mathbb{R}^3)\to\mathbb{R}$ defined by

restricted to the sphere $S_{m}$ which is defined in (1.8). Set

We will summarize some properties of $E_{m}$ under our assumptions. Before the proof, we introduce the well-known Gagliardo-Nirenberg inequality ^[38], which is very useful in the subsequent proof, for some positive constant $C(l)$,

where $\beta_l = \frac{3}{2}-\frac{3}{l}$ and $l\in[2, 6]$.

As in [36, Lemma 2.2], we have the following result.

Lemma 2.1. Assume that $(f_1)-(f_3)$ are satisfied. Then, the following conclusions hold.

${({\rm{i}})}$ For any $m > 0$, $I$ is coercive and bounded from below on $S_{m}$, and, thus, $E_m$ is well-defined. Moreover, $E_m\leq0$.

${({\rm{ii}})}$ There exists $m_0 > 0$ such that $E_m < 0$ for any $m > m_0$.

${({\rm{iii}})}$ If $(f_4)$ holds, then one has $E_m < 0$ for any $m > 0$.

${({\rm{iv}})}$ If $(f'_4)$ holds, then one has $E_m = 0$ for $m > 0$ small enough.

${({\rm{v}})}$ The function $m\to E_m$ is continuous and nonincreasing.

Proof. The proof can be found in [36, Lemma 2.2]. For the reader's convenience, we state their proofs.

(ⅰ) Note that $(f_1)$ and $(f_2)$ imply that for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

Then, for any $u\in H_r^1(\mathbb{R}^3)$, from (2.4) and (2.3), we deduce that

Then, choosing $\varepsilon = \frac{b}{8C_{\frac{14}{3}}m^\frac{1}{3}}$, for $u\in S_{m}$, we have

which implies $I_\mathcal{V}$ is coercive and bounded from below on $S_{m}$, and, thus, $E_m$ is well-defined.

For any $u\in H_r^1(\mathbb{R}^3)$ and $s\in\mathbb{R}$, we define $(s*u)(x): = e^{3s/2}u(e^s x)$ for a.e. $x\in\mathbb{R}^3$. Fixing $u\in S_{m}\cap L^\infty(\mathbb{R}^3)$, it is clear that $s*u\in S_{m}$ and

Then, by $(f_1)$, we derive that

Thus, $E_m\leq0$ for any $m > 0$.

(ⅱ) In view of $(f_3)$ and arguing as in [39, Theorem 2], we can find a function $u\in H_r^1(\mathbb{R}^3)$ such that $\int_{\mathbb{R}^3}F(u)dx > 0$. For any $m > 0$, we set $u_m(x): = u\left(\left(\frac{\|u\|^2_2}{m}\right)^{\frac{1}{3}}x\right)$. Clearly, $u_m\in S_{m}$. Then, it follows that

which implies that $E_m\leq I(u_m) < 0$ for $m > 0$ large enough.

(ⅲ) For any $m > 0$, we choose $u\in S_m\cap L^\infty(\mathbb{R}^3)$. By $(f_3)$, for $M: = \frac{a\|\nabla u\|_2^2}{\|u\|_{\frac{10}{3}}^{\frac{10}{3}}} > 0,$ there exists $\delta > 0$ such that $F(t)\geq M|t|^{\frac{10}{3}}$ for any $|t|\leq \delta$. Then, for any $s < 0$ small enough such that $\|s*u\|_\infty\leq \delta$ and $e^{2s}\|\nabla u\|_2^2 < \frac{2a}{b}$, we have

as required.

(ⅳ) Fix $p\in(\frac{10}{3}, \frac{14}{3})$. By $(f_2)$ and $(f'_4)$, there exists $C > 0$ such that

For any $u\in H_r^1(\mathbb{R}^3)$, from (2.3), we have

Taking $m$ small enough such that

for any $u\in S_m$, by (2.6), we conclude that

By Young's inequality and (2.8), one has

and if we choose $m > 0$, it satisfies

Therefore, from (2.8) and (2.9), we deduce $I(u)\geq0$ for any $u\in S_{m}$ if we choose $m > 0$ small enough such that (2.7) and (2.10) hold. Therefore, from (i), we infer that $E_m = 0$ for $m > 0$ small enough.

(ⅴ) To show the continuity, it is equivalent to prove that for a given $m > 0$, and any positive sequence ${m_k}$ such that $m_k\rightarrow m$ as $k\rightarrow \infty$, one has

In view of the definition of $E_{m_k}$, for every $k\in\mathbb{N}$, let $u_k\in S_{m_k}$ such that

From (2.5), it follows that $\{u_k\}$ is bounded in $H_r^1(\mathbb{R}^3)$. Noting that $\sqrt{\frac{m}{m_k}}u_k\in S_{m}$, from $m_k\to m$ as $k\rightarrow \infty$, (2.4), and (2.12), similar to the proof of [40, Lemma 2.4], we obtain that

On the other hand, choosing a minimization sequence $\{v_n\}\in S_{m}$ for $I$, we can follow the same line as in (2.13) to obtain that $E_{m_k}\leq E_m+o(1)$. Therefore, we obtain (2.11).

To show that $E_m$ is nonincreasing in $m > 0$, we first claim that for any $m > 0$,

Indeed, for any $u\in S_{m}$ and $t > 1$, set $v(x): = u(t^{-\frac{1}{3}}x)$. Then, $v\in S_{tm}$ and we deduce that

Since $u\in S_{m}$ is arbitrary, we obtain the inequality (2.14). As a consequence, from (ⅰ) and (2.14), it follows that $E_m$ is nonincreasing.

In view of Lemma 2.1,

is well-defined and it is easy to obtain the following property of $m^*$.

Lemma 2.2. Assume that $(f_1)-(f_3)$. Then, the following statements are true.

${({\rm{i}})}$ If $(f_4)$ holds, then $m^* = 0$.

${({\rm{ii}})}$ If ($f'_4)$ holds, then $m^* > 0$; in addition, $E_{m} = 0$ for $m\in(0, m^*]$ and $E_{m} < 0$ for $m\in(m^*, +\infty)$.

Noting that $E_{m} < 0$ for any $m > m_{\ast}$ and $E_{\mathcal{V}, m} = E_{m}+\frac{\mathcal{V}m}{2}$, an immediate consequence of Lemma 2.2 is the following corollary.

Corollary 2.3. Assume that $f$ satisfies the conditions $(f_1)-(f_3)$. Then, the following conclusions hold.

${({\rm{i}})}$ For any $m > 0$, $I_\mathcal{V}$ is coercive and bounded from below on $S_{m}$, and, thus, $E_{\mathcal{V}, m}$ is well-defined.

${({\rm{ii}})}$ If $(f_4)$ or $(f'_4)$ holds, then, for any $m > m^*$, there exists $\Theta_m > 0$ such that $E_{\mathcal{V}, m} < 0$ for any $0\le\mathcal{V}\leq\Theta_m$.

Lemma 2.4. Assume that $f$ satisfies the conditions $(f_1)-(f_3)$, and either $(f_4)$ or $(f'_4)$ holds. Then, for any $m > m^*$, fix $\mathcal{V}\in[0, \Theta_m]$, where $\Theta_m$ is defined in Corollary 2.3, we have $\frac{k}{m}E_{\mathcal{V}, m} < E_{\mathcal{V}, k}$ for all $k\in(0, m)$.

Proof. If ($f'_4)$ holds, from Lemma 2.2 (ⅱ), we conclude that $E_{\mathcal{V}, k} = \frac{\mathcal{V}k}{2} > 0$ for all $k\in(0, m^*]$, which implies $\frac{k}{m}E_{\mathcal{V}, m} < 0 < E_{\mathcal{V}, k}$ for all $k\in(0, m^*]$. Therefore, either $(f_4)$ or $(f'_4)$ holds, and we assume $k\in(m^*, m)$.

Let $t = \frac{m}{k}$ and $\{u_n\}\subset S_{k}$ such that $I_{\mathcal{V}}(u_n)\to E_{\mathcal{V}, k}$. We claim that there exists $C > 0$ such that

Indeed, if (2.17) is not true, then passing to a subsequence, $\left\|\nabla u_n\right\|^2_2\rightarrow 0$. Then, by (2.4) and (2.3), we obtain

Then, recalling $k > m^*$, by Corollary 2.3, we deduce that for $\mathcal{V}\in[0, \Theta_m]$,

a contradiction.

Since $t > 1$, noting that $v_n(x) = :u_n(t^{-\frac{1}{3}}x)\in S_{m}$, from (2.17), we deduce that

which implies

The proof is complete.

In the following, we always assume that $f$ satisfies the conditions $(f_1)-(f_3)$ and either $(f_4)$ or $(f'_4)$ holds. Now, we give the following compactness result for $I_{\mathcal{V}}$ on $S_{m}$, which will play a crucial role in our proof.

Lemma 2.5. Fix $m > m^*$. Let $\mathcal{V}\in[0, \Theta_m]$, and $\{u_n\}\subset S_m$ be a minimizing sequence with respect to $I_{\mathcal{V}}$. Then, $\{u_n\}$ has a strongly convergent subsequence.

Proof. Since $E_{\mathcal{V}}$ is coercive on $S_{m}$, the sequence $\{u_n\}$ is bounded. Then, up to a subsequence, there exists $u\in H_r^1(\mathbb{R}^3)$ such that $u_n\rightharpoonup u$. Moreover, since $H_r^1(\mathbb{R}^3)\hookrightarrow L^q(\mathbb{R}^3)$ $(2 < q < 6)$ is compact, one has $u_n\to u$ in $L^q(\mathbb{R}^3)$ $(2 < q < 6)$.

To begin, we suppose $u\ne0$ and $\|u\|_2^2 = \bar{m} < m$. Set

By the Brezis-Lieb Lemma (see ^[37]),

we infer that $m = \bar{m}+d$ and $\bar{m}$, $d_n\in(0, m)$ for $n$ large enough. Furthermore, it follows from (2.4) and $u_n\to u$ in $L^q(\mathbb{R}^3)$ $(2 < q < 6)$ that

Hence, from Lemma 2.4, we deduce that

Letting $n\to+\infty$ and using Lemma 2.4 again, we derive that

a contradiction. Thus, we get $\|u\|_2^2 = m$. Consequently,

Noticing $E_{\mathcal{V}, m} = \mathop{\lim}\limits_{n\to+\infty}I_{\mathcal{V}}(u_n)$, $\|u\|_2^2 = m$ and (2.19), we conclude that

As a consequence,

Therefore, from (2.20) and (2.19), we deduce that

which ensures $u_n\to u$ in $H_r^1(\mathbb{R}^3)$.

Now, we assume $u = 0$. Clearly, by (2.4) and $u_n\to 0$ in $L^q(\mathbb{R}^3)$ $(2 < q < 6)$, we get

which implies

a contradiction. This proves the lemma.

Theorem 2.6. Fix $m > m^*$. Then, for any $\mathcal{V}\in[0, \Theta_m]$, problem (2.1) has a couple $(u, \mu)$ solutions, where $u$ is positive, radial, and $\mu < 0$.

Proof. We divide our proof into two steps.

$Step\; 1.$ By Lemma 2.3 and Lemma 2.5, there exists a bounded minimizing sequence $\{u_n\}\subset S_{m}$ with respect to $E_{\mathcal{V}, m}$ and $u\in S_{m}$ such that $u_n\to u$ in $H_r^1(\mathbb{R}^3)$ and $I_{\mathcal{V}}(u_n)\to E_{\mathcal{V}, m} = I_{\mathcal{V}}(u)$. We define $\psi:H_r^1(\mathbb{R}^3)\to\mathbb{R}$ by

From the Lagrange multiplier, there exists $\mu\in\mathbb{R}$ such that

where $(H_r^{1}(\mathbb{R}^3))^*$ denotes the dual space of $H_r^1(\mathbb{R}^3)$. Hence,

Therefore,

that is, $\mu < 0$.

$Step\; 2.$ By the fact $I_{\mathcal{V}}(u) = E_{\mathcal{V}, m}$ and the definition of the functional $I_{\mathcal{V}}$ and $S_{m}$, clearly, $I_{\mathcal{V}}(|u|) = I_{\mathcal{V}}(u) = E_{\mathcal{V}, m}$ and $|u|\in S_{m}$. So, we can replace $u$ by $|u|$.

Now we prove $u$ is positive. Arguing indirectly, we assume that there exists $x_0\in\mathbb{R}^3$ such that $u(x_0) = 0$. Since $u\ne0$, there exists $x_1\in\mathbb{R}^3$ such that $u(x_1) > 0$. Fix $R_1 > 0$ large enough such that $x_0, x_1\in B_{R_1}(0)$. By the Harnack inequality [41, Theorem 8.20], there exists $C_5 > 0$ such that

Combining this, $u(x_1) > 0$, and $u(x_0) = 0$, we get the contradiction from

The proof is complete.

From the result of Theorem 2.6, we can get the following corollary:

Corollary 2.7. Fix $m > m^*$ and let $0\le\mathcal{V}_1 < \mathcal{V}_2\le \Theta_m$. Then, $E_{\mathcal{V}_1, m} < E_{\mathcal{V}_2, m} < 0$.

Proof. Let $u\in S_m$ satisfy $I_{\mathcal{V}_2}(u) = E_{\mathcal{V}_2, m}$. Then, $E_{\mathcal{V}_1, m}\le I_{\mathcal{V}_1}(u) < I_{\mathcal{V}_2}(u) = E_{\mathcal{V}_2, m} < 0.$

Remark 2.1. We denote $E_{0, m}$ and $E_{\infty, m}$ by the following real numbers:

An immediate result of Corollary 2.7 and condition $(V)$ is

3.   Proof of Theorem 1.1

In this section, we will prove our main result. From now on, we always assume that $f$ is odd and satisfies the conditions $(f_1)-(f_3)$. Moreover, we assume that either $(f_4)$ or $(f'_4)$ holds, $m > m^*$, and $\|V\|_\infty\leq\Theta_m$, where $m^*$ and $\Theta_m$ are given in (2.16) and Corollary 2.3, respectively.

To start, we manage to study the convergence of the Palais-Smale sequence for $I_\varepsilon$ at some negative level. Denote

From Remark 2.1, we fix $0 < \rho = \frac{1}{2}\left(E_{\infty, m}-E_{0, m}\right)$. We need following result which describes the relation between the levels $E_{\varepsilon, m}$ and $E_{\infty, m}$ playing an important role in our proof.

Lemma 3.1. There is $\varepsilon_0 > 0$ such that $E_{\varepsilon, m} < E_{\infty, m}$ for all $\varepsilon\in\left(0, \varepsilon_0\right)$.

Proof. Let $u_0\in S_{m}$ with $I_{0}(u_0) = E_{0, m}$. Then, by the definition of $E_{\varepsilon, m}$,

Therefore,

From this and Remark 2.1, the estimate $E_{\varepsilon, m} < E_{\infty, m}$ is established for $\varepsilon$ small enough.

From now on, fix $\varepsilon\in(0, \varepsilon_0)$, where $\varepsilon_0$ is given in Lemma 3.1.

Lemma 3.2. If $\{u_n\}\subset S_{m}$ satisfies $I_\varepsilon(u_n)\to c$ with $c < E_{0, m}+\rho < 0$ and $u_n\rightharpoonup u$ in $H_r^1(\mathbb{R}^3)$, then, $u\ne0$.

Proof. Assume by contradiction that $u = 0$. By $(V)$, for any given $\alpha > 0$, there is $R > 0$ such that for any $|x|\geq R$,

Based on assumptions and the boundedness of $\{u_n\}$ in $H_r^1(\mathbb{R}^3)$ and $u_n\to0$ in $L_{loc}^2\left(\mathbb{R}^3\right)$, we deduce that for some $C > 0$,

Due to the arbitrariness of $\alpha$, we have $E_{0, m}+\rho\ge E_{\infty, m}$, which is absurd. The proof is complete.

Lemma 3.3. For each $\varepsilon\in(0, \varepsilon_0)$, the functional $I_\varepsilon$ satisfies the $(PS)_c$ condition restricted to $S_{m}$ for $c < E_{0, m}+\rho$, namely, if any sequence $\{u_n\}\subset S_{m}$ such that

then $\{u_n\}$ has a convergent subsequence.

Proof. Similar to Corollary 2.3 (ⅰ), we can verify that $I_\varepsilon$ is coercive on $S_{m}$. Thus, $\{u_n\}$ is bounded in $H_r^{1}(\mathbb{R}^3)$. Up to a subsequence, we assume that $u_n\rightharpoonup u_\varepsilon$ in $H^{1}(\mathbb{R}^3)$ and $u_n\to u_\varepsilon$ in $L^q(\mathbb{R}^3)$ ($2 < q < 6$). From Lemma 3.2, $u_\varepsilon\neq0$.

We define $\psi:H_r^1(\mathbb{R}^3)\to\mathbb{R}$ by

Then, by Willem [37, Prosition 5.12], there exists $\{\mu_n\}\subset\mathbb{R}$ such that

From $\psi'(u_n)u_n = m$, the boundedness of $\{u_n\}$ in $H_r^1(\mathbb{R}^3)$, $(f_1)$, $(f_2)$, and (2.3), it follows that

where $C_1, C_2$ are positive constants independent of $\varepsilon$ and $n$. This implies that $\{\mu_n\}$ is a bounded sequence. As a consequence, we assume, up to a subsequence, $\mu_n\to\mu_\varepsilon$ as $n\to+\infty$, and, thereby,

Meanwhile,

where $A_\varepsilon = \lim_{n\to\infty}\int_{\mathbb{R}^{3}}|\nabla u_n|^2dx\ge0$.

Now, we verify that there exists $\mu_\ast < 0$ independent of $\varepsilon$ such that

Noting that $u_n\rightharpoonup u_\varepsilon$ in $H_r^1(\mathbb{R}^3)$, one has $u_n\to u_\varepsilon$ in $L^q(\mathbb{R}^3)$ $(2 < q < 6)$. Then, by (2.4), we conclude that

Therefore, from (3.3), we deduce that

which implies $\mu_\varepsilon < 0$. In addition, we also have

Therefore,

Hence, (3.4) holds.

Now we prove that $u_n\to u_\varepsilon$ in $H_r^1(\mathbb{R}^3)$. Set $v_n: = u_n-u_\varepsilon$. From (3.2) and (3.3), we infer that

Then, using (3.4), (2.4), and the fact that $v_n\to0$ in $L^q(\mathbb{R}^3)$ ($2 < q < 6$), we conclude that

where positive constant $C$ does not rely on $\varepsilon$. Thus, $v_n\to 0$ in $H_r^1(\mathbb{R}^3)$, i.e., $u_n\to u_\varepsilon$ in $H_r^1(\mathbb{R}^3)$. Therefore, $\|u_\varepsilon\|^2_2 = m$ and

The proof is finished.

Fix $\delta > 0$ and define $\eta$ as a smooth nonincreasing cutoff function in $[0, +\infty)$ by

Recall that $M = \{x\in\mathbb{R}^3:V(x) = 0\}$. For any $y\in M$, let us define

where $w_0$ is a positive radial solution of the problem

with $I_0(w_0) = E_{0, m}$. Then, let

and denote $\Phi_\varepsilon: M\to S_{m}$ by $\Phi_\varepsilon(y) = \tilde{\phi}_{\varepsilon, y}$. Obviously, it has compact support for any $y\in M$. In addition, let $R = R(\delta) > 0$ be such that $M_{\delta}\subset B_R(0)$. Define $\chi:\mathbb{R}^3\to\mathbb{R}^3$ as

Finally, let us consider $\omega_\varepsilon:S_{m}\to\mathbb{R}^3$ given by

Lemma 3.4. The function $\Phi_\varepsilon$ has the following two limits:

$(1)$ $\mathop{\lim}\limits_{\varepsilon\to0}I_\varepsilon\left(\Phi_\varepsilon(y)\right) = E_{0, m}, \; \; \; \mathit{\text{uniformly in}}\; \; \; y\in M,$

$(2)$ $\mathop{\lim}\limits_{\varepsilon\to0}\omega_\varepsilon\left(\Phi_\varepsilon(y)\right) = y, \; \; \; \mathit{\text{uniformly in}}\; \; \; y\in M.$

Proof. $(1)$ Assume that $\{y_n\}\subset M$. From Lebesgue's dominated convergence theorem, it follows that

Likewise, we also have

and

Consequently,

and this proves the first limit.

$(2)$ Suppose by contradiction that there is $\delta_0 > 0$, $\{y_n\}\subset M$ with $y_n\to y\in M$ and $\varepsilon\to0$ such that

Using the definitions of $\Phi_{\varepsilon_n}(y_n)$ and $\omega_{\varepsilon_n}$, combined with $\{y_n\}\subset M\subset B_R(0)$ and Lebesgue's dominated convergence theorem, we deduce that

which contradicts (3.7), and this proves the desired result.

Let $\gamma:[0, +\infty)\to[0, +\infty)$ be a positive function such that $\gamma(\varepsilon)\to0$ as $\varepsilon\to0$, then define $\tilde{S}_{m}$ as

Thanks to $(1)$ of Lemma 3.4, the function

satisfies $\gamma(\varepsilon)\to0$ as $\varepsilon\to0$. Hence, $\Phi_\varepsilon(y)\in\tilde{S}_{m}$ for all $y\in M$.

Proposition 3.5. Let $\varepsilon_n\to0$ and $\{u_n\}\subset S_{m}$ with $I_{\varepsilon_n}(u_n)\to E_{0, m}$. Then, $\{u_n\}$ has a convergent subsequence in $H_r^1(\mathbb{R}^3)$.

Proof. Since $\{u_n\}\subset S_m$, from $(V)$, we deduce that

which implies $I_0(u_n)\to E_{0, m}$ as $n\to+\infty$. From Lemma 2.5, $\{u_n\}$ has a convergent subsequence in $H_r^1(\mathbb{R}^3)$.

Lemma 3.6.

Proof. Let $\varepsilon_n\to0$ and $u_n\in\tilde{S}_{m}$ such that

Since $u_n\in\tilde{S}_{m}$, by the definition of $\tilde{S}_{m}$, we deduce that $u_n\in S_{m}$, and as $\varepsilon_n\to0$,

from which it follows that $I_{\varepsilon_n}(u_n)\to E_{0, m}$. From Proposition 3.5, $\{u_n\}$ is strongly convergent to some $u\in H_r^1(\mathbb{R}^3)$. Then, due to the definition of $\omega_{\varepsilon_n}$ and $u_n\in S_{m}$, using the Lebesgue's dominated convergence theorem, we obtain that

that is, $\omega_{\varepsilon_n}(u_n) = o_n(1)$. Noting that $0\in M$, we conclude that

The proof is complete.

Proof of Theorem 1.1. We will divide the proof into two parts:

$Step\; \; 1$: Multiplicity of solutions.

Set $\varepsilon\in(0, \varepsilon_0)$ and fix $\delta > 0$. By Lemmas 3.4 and 3.6, we can obtain that the diagram $M\overset{\Phi_\varepsilon}{\rightarrow }S_{m}\overset{\omega_\varepsilon}{\rightarrow }M_\delta$ is well-defined. For $\varepsilon$ small enough, we denote $\omega_\varepsilon(\Phi_\varepsilon(y)): = y+\varsigma(y)$ for $y\in M$ and $Q(t, y): = y+(1-t)\varsigma(y)$. By Lemma 3.7, $\|\varsigma(y)\|\leq\frac{\delta}{2}$ uniformly for $y\in M$. Obviously, the continuous function $Q:[0, 1]\times M\to M_\delta$ satisfies $Q(0, y) = \omega_\varepsilon(\Phi_\varepsilon(y))$ and $Q(1, y) = y$ for any $y\in M$. Therefore, $\omega_\varepsilon\circ\Phi_\varepsilon$ is homotopic to the inclusion map $id:M\to M_\delta$. In view of ^[42], we arrive at

Recall that $I_\varepsilon$ is bounded from below on $S_{m}$. Moreover, from Lemma 3.3, $I_\varepsilon$ satisfies the $(PS)_c$ condition for $c\in\left(E_{0, m}, E_{0, m}+\gamma(\varepsilon)\right)$. Then, due to the Lusternik-Schnirelmann category of critical points (see ^[37,43]), we infer that $I_\varepsilon$ admits at least $cat_{M_\delta}(M)$ critical points on $S_{m}$.

$Step\; \; 2$: Concentration phenomena of the solutions.

Let $u_\varepsilon$ be a solution of (1.7) with $I_\varepsilon(u_\varepsilon)\le E_{0, m}+\gamma(\varepsilon)$, where $\gamma$ was given in (3.8). From Proposition 3.5, for any $\varepsilon_n\to0$, there exists $u\in H_r^1(\mathbb{R}^3)\setminus\{0\}$ such that $u_{\varepsilon_n}\to u$ in $H_r^1(\mathbb{R}^3)$. Clearly, as in (3.5), $u_n: = u_{\varepsilon_n}$ satisfies

with

Since $u_{n}\to u$ in $H_r^1(\mathbb{R}^3)$, similar to [8, Lemma 4.5], we obtain

As a consequence, given $\theta > 0$, there exist $R > 0$ and $n_{0}\in\mathbb{N}$ such that

for $|x|\ge R$ and $n\ge n_{0}$. We claim that $\|u_n\|_\infty\nrightarrow0$; otherwise we will have $u_n\to0$ in $H_r^1(\mathbb{R}^3)$, contrary to $u\neq0$. Now, we fix $\theta > 0$ small such that $\|u_n\|_\infty\ge2\theta$ and choose $\xi_n\in\mathbb{R}^3$ such that $|u_n(\xi_n)| = \|u_n\|_\infty$ for all $n\in\mathbb{N}$. It follows that $|\xi_n|\le R$ for all $n\in\mathbb{N}$. Therefore,

as required.

Author contributions

Yangyu Ni: Writing-original draft, Writing-review & editing; Jijiang Sun: Supervision, Writing-review & editing, Methodology, Validation; Jianhua Chen: Formal Analysis, Validation.

Use of AI tools declaration

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

Acknowledgments

J. Sun is supported by NSFC (No.12361024) and Jiangxi Provincial Natural Science Foundation (No.20232ACB211004), J. Chen is supported by Jiangxi Provincial Natural Science Foundation (No.20232BAB201001).

Conflict of interest

The authors declare there is no conflict of interest.