Normalized solutions to nonautonomous Kirchhoff equation

1.   Introduction

In this paper, we consider the existence of solutions with prescribed $L^{2}$-norm to the following Kirchhoff problem with a perturbation

where $1\le N\le 3, a, b, c > 0, p\in (2, 2^{*}), q \in [1, 2)$, $h(x):\mathbb{R}^{N}\to \mathbb{R}$ is a potential, $2^{*} = 6$ if $N = 3$, and $2^{*} = +\infty$ if $N = 1, 2$. Based on these observations, we establish the existence of normalized solutions under different assumptions on $h(x)$.

The energy functional of Eq.(1.1) is defined by

constrained on the $L^{2}$-spheres in $H^{1}(\mathbb{R}^{N})$:

In 1883, Kirchhoff ^[1] first proposed the following nonlinear wave equation

which extends the original wave equation by describing the transversal oscillations of a stretched string and, particularly, by considering the subsequent change in string length caused by oscillations. Thereafter, there was a boom in the study of the Kirchhoff-type equation. We can refer to ^[2,3,4] for the physical background about Kirchhoff problem.

Mathematically, Eq.(1.1) is not a pointwise identity as a result of the emergence of the term $(b\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx)\Delta u$. This causes some mathematical difficulties. In the renowned paper ^[5], J.L. Lions raised an abstract framework that has received much attention. There are two ways to study the Kirchhoff-type equation. The first approach is to consider fixing the parameter $\lambda \in \mathbb{R}$. In this case, there are a lot of results, which have been widely studied by using variational methods. We can refer to ^[6,7,8,9] and the references therein. Another way is to fix the $L^{2}$-norm. In this case, the desired solutions have a priori prescribed $L^{2}$-norm, which are usually referred to as normalized solutions in the literature; that is, for any fixed $c > 0$, we take $(u_{c}, \lambda_{c})\in H^{1}(\mathbb{R}^{N})\times \mathbb{R}$ as a normalized solution with $\|u_{c}\|_{2}^{2} = c$, $\lambda_{c}$ is a Lagrange multiplier. From a physical perspective, the $L^{2}$-prescribed norm represents the number of particles of each component in Bose-Einstein condensates or the power supply in a nonlinear optics framework. In addition, the $L^{2}$-prescribed norm can provide a better insight on the dynamical properties, like orbital stability or instability, and can describe attractive Bose-Einstein condensates.

For the local case, i.e., $b = 0$, Eq.(1.1) reduces to the general Schrödinger type:

which dates back to the groundbreaking work by Stuart. In ^[10], Stuart tackled the problem (1.3) for $f(x, u) = |u|^{p-2}u$ and $p\in(2, 2+\frac{4}{N})$ ($L^2$-subcritical case); here, $2+\frac{4}{N}$ is called the $L^2$-critical exponent. For $L^2$-subcritical case, the minimization method is the conventional method to find normalized solutions. When $f$ is $L^2$-supercritical growth, a groundbreaking work in the $L^{2}$-supercritical case was accomplished by Jeanjean^[11]. Jeanjean developed a novel argument related to the mountain pass geometry by the stretched functional. Bartsch and Soave ^[12,13] also proposed a new approach by using a minimax principle based on the homotopy stable family to prove the existence of normalized solutions for the problem (1.3). Moreover, Soave in ^[14] studied the combined nonlinearity case $f(x, u) = |u|^{p-2}u+\mu |u|^{q-2}u$, $2 < q\leq 2+\frac{4}{N}\leq p < 2^*$ and $q < p$, where $2^* = \infty$ if $N\leq2$ and $2^* = \frac{2N}{N-2}$ if $N\geq3$. Soave showed that nonlinear terms with different power strongly affects the geometry of the functional and the existence and properties of ground states.

When $f(x, u) = a(x)f(u)$, the solutions to the nonautonomous problem were first studied by Chen and Tang ^[15]. Compared with the autonomous problems, the main challenge of the problem is constructing a $(PS)$ sequence with an additional property to recover the compactness. Very recently, Chen and Zou ^[16] studied the following problem with a perturbation

where $h(x)\ge 0$. For $p\in(2, 2+\frac{4}{N})$ and an arbitrarily positive perturbation, Chen and Zou proved that there exists a global minimizer with negative energy. The existence of a mountain pass solution with positive energy for $p\in(2+\frac{4}{N}, 2^{*})$ was studied. We can refer to ^[17,18,19] for more details.

For the nonlocal case, i.e., $b > 0$, the more general form of Eq.(1.1) is the following equation

which has attracted considerable attention. When $f(x, u) = |u|^{p-2}u$ (i.e., the limited problem of Eq.(1.1)), the problem (1.5) turns to

where $a, b, c > 0$ are constants, $1\le N\le 3$, and $p\in (2, 2^{*})$. The energy functional of (1.6) is

By the Gagliardo-Nirenberg inequality ^[20] for any $p\in (2, 2^{*})$

where $\gamma_{p} = \frac{N(p-2)}{2p},$ we can get $L^{2}$-critical exponent $\bar{p} = 2+\frac{8}{N}$ of the Kirchhoff problem. It is well known that Ye ^[21] obtained the sharp existence of global constraint minimizers for Eq.(1.6) in the case of $p\in (2, \bar{p})$. When $p\in (2+\frac{4}{N}, \bar{p})$, Ye proved a local minimizer, which is a critical point of $\left.I_{\infty}\right|_{S_{c}}$. By considering a global minimization problem

we have

for any given $c > 0$. We can see that the minimization method is not feasible for $p\in (\bar{p}, 2^{*})$. Then, Ye proved the existence of normalized solutions by taking advantage of the Pohozaev constraint method in the case of $p\in (\bar{p}, 2^{*}).$ For the $L^{2}$-critical case of $\bar{p} = 2+\frac{8}{N}$, Ye ^[22] showed the existence and mass concentration of critical points. Using some simple energy estimates instead of the concentration-compactness principles introduced in ^[21], Zeng studied the existence and uniqueness of normalized solutions for $p\in (2, 2^{*})$ in ^[23].

Additionally, Li, Luo, and Yang ^[24] proved the existence and asymptotic properties of solutions to the following equation with combined nonlinearity

where $a, b, c, \mu > 0$, $2 < q < \frac{14}{3} < p \le 6$ or $\frac{14}{3} < q < p \le 6$. They showed a multiplicity result for the case of $2 < q < \frac{10}{3}$ and $\frac{14}{3} < p < 6$ and obtained the existence of ground state normalized solutions for $2 < q < \frac{10}{3} < p = 6$ or $\frac{14}{3} < q < p\le 6$. They also showed some asymptotic results on the obtained solutions. For the case $\mu \le0$, in ^[25], Carri$\tilde{a}$o, Miyagaki, and Vicente studied the ground states existence of Eq.(1.11) for $2 < q < 2^{*}, p = 2^{*}$ or $2 < q \le \bar{p} < p < 2^{*}$. For the nonautonomous problem, when $f(x, u) = |u|^{p-2}u+V(x)|u|^{q-2}u$, $N = 3$, $p = \frac{14}{3},$ $q = 4$ and $V \in L_{loc}^{\infty}(\mathbb{R}^{3})$, Ye ^[26] considered the existence of minimizers to the nonautonomous problem. Moreover, $V(x)$ satisfies

By the concentration compactness principle, if $b < b_{0}$, Ye showed that there exists $a_{0}, c_{0} > 0$ such that the above problem has a minimizer for all $a < a_{0}$ and $c < c_{0}$. Additionally, when $f(x, u) = K(x)f(u)$, Chen and Tang ^[27] considered the existence of ground state solutions, where $K(x) \in C(\mathbb{R}^{3}, \mathbb{R}^{+})$ and $f(u)$ is $L^{2}$-supercritical. When $2+\frac{4}{N} < p < 2+\frac{8}{N}$, the geometric structure of the energy functional is more complex, especially when $h(x) > 0$, and there are very few works studying this range with potential. Other results about normalized solutions of Kirchhoff equation in a more general form can be found in ^{[28,29,30,31]}.

Motivated by the results above, when $\mu$ of Eq.(1.11) is replaced by a potential function $h(x)$ and $1\le q < 2$, there are no results in studying normalized solutions of such nonautonomous Kirchhoff equations with a small perturbation. In the present paper, we first obtain the normalized solution of this type of equation, which can be seen as an extension of some known results in the literature.

Let us now outline the main strategy to prove the three results of this paper under different assumptions on $h(x)$. First, we treat the mass-subcritical case $2 < p < 2+\frac{4}{N}$: for any $c > 0$, we set

It is standard that the minimizers of $l_{c}$ are critical points of $\left.I\right|_{S_{c}}$. We introduce the following assumptions on $h(x)$.

Now we state the main results of this paper:

Theorem 1.1. Suppose $1\le N\le3$, $2 < p < 2+\frac{4}{N}$ and $h(x)\ge 0$ satisfies $(\mathbf{h}_{\mathbf{1}})$. Then, for all $c > 0$, $l_{c}$ has a minimizer, hence Eq.(1.1) has a normalized ground state solution.

Remark 1.1. Notice that the minimizer obtained in Theorem 1.1 is a global minimizer rather than a local minimizer. It is easy to find that the energy functional is coercive on $S_{c}$, which hints that each minimizing sequence $\{u_{n}\}$ is bounded on $S_{c}$. The main difficulty of proof is to show that the minimizing sequence $\{u_{n}\}$ converges strongly to $u\ne 0$ in $H^{1}(\mathbb{R}^{N})$. The key step is to establish the inequality $l_{c_{1}+c_{2}}\leq l_{c_{1}}+l_{\infty, c_{2}}$ for $c_{1}, c_{2} > 0$ (see Lemma 2.2), which is crucial to recover the compactness.

Next, while addressing the $L^{2}$-supercritical case, the functional is unbounded from below on $S_{c}$, thus the minimizing approach on $S_{c}$ is not valid anymore. Ye ^[21] proved that $l_{\infty, c} = -\infty$ for all $c > 0$ if $p\in \left(2+\frac{8}{N}, 2^{*}\right)$, and proved the existence of one normalized solution by a suitable submanifold of $S_{c}$. In this paper, after the appearance of a very small perturbation term, we want to show that the energy functional $I$ has a mountain pass geometry and show the existence of a mountain pass solution with positive energy level for $p\in \left(2+\frac{8}{N}, 2^{*}\right)$. We require the perturbation $h(x)$ to have a higher regularity. We need to assume that:

We have the following result.

Theorem 1.2. Suppose $1\le N\le3$, $2+\frac{8}{N} < p < 2^{*}$ and $h(x)$ satisfies $(\mathbf{h}_{\mathbf{2}})$. Let $c > 0$ be fixed. Moreover,

Then, Eq.(1.1) has a mountain pass solution $u$ at a positive energy level.

Remark 1.2. We are going to use the minimax characterization to find a critical point. Although the mountain pass geometry of the functional on $S_{c}$ can be obtained easily, unfortunately the boundedness of the obtained $(PS)$ sequence is not yet clear. In this paper, we adopt a similar idea to ^[11] and construct an auxiliary map $\tilde{I}(t, u): = I(t\star u),$ which on $\mathbb{R}\times S_{c}$ has the same type of geometric structure as $I$ on $S_{c}$. Besides, the $(PS)$ sequence of $I$ satisfies the additional condition (see Lemma 3.5), which is the key ingredient to obtain the boundedness of the $(PS)$ sequence.

Finally, we will discuss $h(x)\le0$, and the problem becomes more delicate and difficult. Although the mountain pass structure by Jeanjean ^[11] is destroyed, Bartsch et al.^[32] established a new variational principle exploiting the Pohozaev identity. For convenience, we define $\bar{h}(x): = -h(x)\ge 0$. Next, we state our basic assumptions on $\bar{h}(x)$.

$(\mathbf{h}_{\mathbf{3}})\ \bar{h}(x)\in L^{\frac{2}{2-q} }(\mathbb{R}^{N})\cap C^{1}(\mathbb{R}^{N}), \ \langle\nabla \bar{h}(x), x\rangle \in L^{\frac{2}{2-q} }(\mathbb{R}^{N})$ and $\bar{h}(x)\ge0$. For some constants $\Upsilon > 0$, $\bar{h}(x)$ satisfies

Theorem 1.3. Assume $1\le N\le3$, $2+\frac{8}{N} < p < 2^{*}$. If $(\mathbf{h}_{\mathbf{3}})$ holds and $\bar{h}(x)$ satisfies

Then Eq.(1.1) has a couple of solutions $(u_, \lambda) \in H^{1}(\mathbb{R}^{N})\times \mathbb{R}$ and $\lambda > 0$.

Remark 1.3. Indeed, when $h(x)\leq0$, the problem is made more difficult by the simultaneous appearance of a negative potential and nonlocal term. We refer to Bartsch et al. ^[32] constructing a suitable linking geometry method to obtain the existence of bound state solutions with high Morse index. The crucial step is to estimate the minimax level $m_{c} < L_{h, c} < 2m_{c}$ (see Lemma 4.3 and Lemma 4.5) to recover the compactness.

Notations: We introduce some notations that will clarify what follows:

$\bullet\ \ H^{1}(\mathbb{R}^{N})$ is the usual Sobolev space with the norm $\|u\| = \left(\int_{\mathbb{R}^{N}}\left|\nabla u\right|^{2}+\left | u \right|^{2} dx \right)^{\frac{1}{2}}$.

$\bullet\ \ L^{p}(\mathbb{R}^{N})$ with $p \in [1, \infty)$ is the Lebesgue space with the norm $\|u\|_{p} = \left (\int_{\mathbb{R}^{N}}\left| u\right|^{p}dx \right)^{\frac{1}{p} }$.

$\bullet$ The arrows $'\rightharpoonup'$ and $'\to'$ denote the weak convergence and strong convergence, respectively.

$\bullet\ \ C, C_{i}$ denote positive constants, which may vary from line to line.

$\bullet\ \ (t\star u)(x): = t^{\frac{N}{2}}u(tx)$ for $t\in \mathbb{R}^{+}$ and $u\in H^{1}(\mathbb{R}^{N})$.

2.   Proof of Theorem 1.1

In this section, for $2 < p < 2+\frac{4}{N}$ and $h(x)\ge0$ we prove Theorem 1.1. By the Gagliardo-Nirenberg inequality (1.8), the Hölder inequality, and the assumption $(\mathbf{h}_{\mathbf{1}})$, we have

thus $I$ is bounded from below on $S_{c}$ since $0 < p\gamma_{p} < 2$.

For $1\le N\le3$ and $2 < p < 2+\frac{4}{N}$, the existence and uniqueness of positive normalized solutions of the limited problem (1.6) have been studied in ^[21]. In order to find the minimizer of $I$ on $S_{c}$, first we state some fundamental properties of $l_{\infty, c}$, which will be crucial to recover the compactness later on. The proof of the next lemma can be found in [28,Theorem 1.1 and Lemma 2.5].

Lemma 2.1. Suppose $1\le N\le3$ and $2 < p < 2+\frac{4}{N}$. Then, for all $c > 0$, we have

(i) the strict sub-additivity for $l_{\infty, c}$, i.e.,

(ii) the limited problem (1.6) has a couple of ground state solutions $(u_{\infty }, \lambda _{c})\in H^{1}(\mathbb{R}^{N}) \times \mathbb{R}$, i.e.,

Next, we introduce the inequality $l_{c_{1}+c_{2}}\leq l_{c_{1}}+l_{\infty, c_{2}}$, which plays a crucial role in proving the convergence of the minimizing sequence.

Lemma 2.2. Suppose $2 < p < 2+\frac{4}{N}$ and $h(x)$ satisfies $(\mathbf{h_{1}})$, then the following holds

(i)$-\infty < l_{c} < l_{\infty, c} < 0$ for $c > 0$;

(ii)$l_{c_{1}+c_{2}}\leq l_{c_{1}}+l_{\infty, c_{2}}$ for $c_{1}, c_{2} > 0$.

Proof. (i) It is obvious that $l_{c} > -\infty$ by (2.1). Moreover, by Lemma 2.1, we have

since $u_{\infty} > 0$ and $h(x)$ satisfies $(\mathbf{h_{1}})$.

(ii) For any $\varepsilon > 0$, $c = c_{1}+c_{2}$, we can find $\varphi_{\varepsilon}, \psi_{\varepsilon} \in \mathcal{C}_{0}^{\infty}(\mathbb{R}^{N})$ such that

Let $u_{\varepsilon, n}(x): = \varphi_{\varepsilon}(x)+\psi_{\varepsilon}(x-n \boldsymbol{e}_{1})$, where $\boldsymbol{e}_{1}$ is the unit vector $(1, 0, \cdots)$ in $\mathbb{R}^{N}$. Since $\varphi_{\varepsilon}$ and $\psi_{\varepsilon}$ have compact support, we see that $u_{\varepsilon, n} \in S_{c}$ and

for large $n$. Moreover, thanks to $h \in L^{\frac{2}{2-q}}(\mathbb{R}^{N})$, we have that $\int_{\mathbb{R}^{N}}h(x)\psi_{\varepsilon}^{q}(x-n \boldsymbol{e}_{1})dx\to 0$ as $n\to \infty$, hence $I(\psi_{\varepsilon}(\cdot -n \boldsymbol{e}_{1}))\to I_{\infty } (\psi_{\varepsilon})$ as $n\to \infty$. It follows that

Passing to the limit, thus $l_{c}\leq l_{c_{1}}+l_{\infty, c_{2}}$ since $\varepsilon > 0$ is arbitrary. □

Let $\{u_{n}\}\subset S_{c}$ be a minimizing sequence for $l_{c}$. By (2.1), we know that $I(u)$ is coercive on $S_{c}$ and deduce that $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$. Thus, there exists a subsequence such that $u_{n}\rightharpoonup u_{0}$ and

We need to prove $I(u_{0}) = l_{c}$ and $\| u _{0}\|_{2}^{2} = c$. Now we argue by contradiction to prove this.

Lemma 2.3. Suppose $2 < p < 2+\frac{4}{N}$ and $h(x)$ satisfies $(\mathbf{h_{1}})$. Then, every minimizing sequence for $l_{c}$ has a strong convergent subsequence in $L^{2}(\mathbb{R}^{N})$.

Proof. We argue by contradiction and assume that $c_{1} < c$. We divide the proof into four steps.

Step 1: There exists $\{y_{n}\}\subset \mathbb{R}^{N}$ and $\mu_{0} \in H^{1}(\mathbb{R}^{N}) \backslash\{0\}$ such that

First, we show by contradiction that

where $B_{1}(y) = \{x \in \mathbb{R}^{N}: |x-y|\leq 1\}$. Suppose, on the contrary, that $\delta_{0} = 0$. Then, $u_{n} \rightarrow u_{0}$ strongly in $L^{p}(\mathbb{R}^{N})$. Since $u_{n} \rightharpoonup u_{0}$ in $H^{1}(\mathbb{R}^{N}), h \in L^{\frac{2}{2-q}}(\mathbb{R}^{N})$, we see that $\int_{\mathbb{R}^{N}} h|u_{n}|^{q} dx \rightarrow \int_{\mathbb{R}^{N}} h |u_{0}|^{q} dx$. Combined with Lemma 2.1 (ii), for $c-c_{1} > 0$, we have that

which is a contradiction with Lemma 2.2 (ii). Therefore, (2.3) holds. From (2.3) and $u_{n} \rightarrow u_{0}$ in $L_{loc}^{2}(\mathbb{R}^{N})$, we can find $\{y_{n}\} \subset \mathbb{R}^{N}$ such that $\int_{B_{1}(y_{n})}\left|u_{n}-u_{0}\right|^{2} dx \rightarrow c_{0} > 0$ and $\left|y_{n}\right| \rightarrow \infty$. Let $u_{n}\left(\cdot+y_{n}\right) \rightharpoonup \mu_{0}$ weakly in $H^{1}(\mathbb{R}^{N})$. Note that $\mu_{0}\neq 0$ since $c_{0} > 0$. Therefore, $\{y_{n}\}$ and $\mu_{0}$ satisfy (2.2). Thus, the proof of Step 1 is complete.

Step 2: We show that $\{y_{n}\}$ and $\left(u_{0}, \mu_{0}\right)$ satisfy

Since $\left|y_{n}\right| \rightarrow \infty$, we have that

According to (2.5), we could let $\delta_{1}: = \lim _{n \rightarrow \infty}\|u_{n}-u_{0}-\mu_{0}(\cdot-y_{n})\|_{2}^{2}$. Then, we have $\delta_{1} = c-c_{1}-c_{2}$, where $c_{2}: = \|\mu_{0}\|_{2}^{2}$. We want to show that $\delta_{1} = 0$. Suppose on the contrary that $\delta_{1} > 0$, by direct calculations we have

From the Brezis-Lieb Lemma, we have

Similarly,

Combining (2.6)–(2.8), we have

Since $u_{n} \rightharpoonup u_{0}$ in $H^{1}(\mathbb{R}^{N})$, $\left|y_{n}\right| \rightarrow \infty$ and $h \in L^{\frac{2}{2-q}}\left(\mathbb{R}^{N}\right)$, we have

Recalling that $l_{\infty, c}$ is continuous with respect to $c > 0$ (see ^[33], Theorem 2.1), we have that

and

Hence by (2.9)–(2.12), we have

However, using Lemma 2.1 (i), for any $c_{2}, \delta_{1} > 0$, there exists $l_{\infty, c_{2}+\delta_{1}} < l_{\infty, c_{2}}+l_{\infty, \delta_{1}}$. Hence, we also have

This gives a contradiction and thus we have that $\delta_{1} = 0$.

Step 3: Moreover, the following holds

and

By (2.9)–(2.12) and $\delta_{1} = 0$, we have that

Combined with Lemma 2.2 (ii), we see that $l_{c} = l_{c_{1}}+l_{\infty, c_{2}}$. $I\left(u_{0}\right) = l_{c_{1}}$ and $I_{\infty}\left(\mu_{0}\right) = l_{\infty, c_{2}}$. Thus, Step 3 is proved.

Step 4: Now, we prove the precompactness of minimizing sequence, i.e., $u _{n}\to u _{0}$ in $L^{2}(\mathbb{R}^{N})$.

We can suppose that $\{u _{n}\}$ are nonnegative. Using the strong maximum principle, we have $u_{0}, \mu_{0} > 0$ and $h(x) > 0$ on a set with positive measure, we have that

Combine with the two following inequalities:

So we have

which is a contradiction. Thus the proof of Lemma 2.3 is completed. □

Proof of Theorem 1.1. From Lemma 2.3, the minimizing sequence $\{u_{n}\}$ satisfies $u_{n}\rightarrow u_{0}$ in $L^{2}(\mathbb{R}^{N})$ and $l_{c} = I(u_{0})$, $c = c_{1}$. Since $\{u_{n}\}\subset S_{c}$ is the minimizing sequence of $l_{c}$, we have $\left.d I\right|_{S_{c}}\left(u_{n}\right) \rightarrow 0$ and there exists a sequence of real numbers $\{\lambda_{n}\}$ such that

for every $\varphi \in H^{1}(\mathbb{R}^{N})$. Hence, by (2.21), we have that

Notice that $h(x)\geq 0$, then by the maximum principle, $u_{0} > 0$, and we finish the proof of Theorem 1.1. □

3.   Proof of Theorem 1.2

In this section, we study the mass-supercritical and Sobolev-subcritical case: $2+\frac{8}{N} < p < 2^{*}$, $1\le N \le 3$, and $h(x)$ satisfies the assumption $(\mathbf{h}_{\mathbf{2}})$. First, we show that the energy functional $I$ possesses a mountain pass geometry, which implies the existence of the $(PS)$ sequence. Next, we prove that the limit of the sequence of the Lagrange multipliers related to the $(PS)$ sequence is positive. Then, by applying the splitting lemma, we recover the compactness for this sequence, which yields the existence of solutions for Eq.(1.1).

In order to study the behavior of $(PS)$ sequence, we introduce the splitting lemma, which plays a crucial role in overcoming the lack of compactness. For $\lambda > 0$, we set

and

Lemma 3.1. Let $\{u_{n}\}\subset H^{1}(\mathbb{R}^{N})$ be a $(PS)$ sequence for $I_{\lambda}$ such that $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{N})$ and $\lim_{n\to \infty}\|\nabla u_{n}\|_{2}^{2} = A^{2}$. Then, there exists an integer $k\geq 0$, $k$ nontrivial solutions $\omega^{1}, \cdots, \omega^{k} \in H^{1}(\mathbb{R}^{N})$ to the following problem

and $k$ sequences $\{y_{n}^{j}\} \subset \mathbb{R}^{N}, 1\leq j \leq k$, such that as $n \rightarrow \infty, |y_{n}^{j}| \rightarrow \infty, |y_{n}^{j_{1}}-y_{n}^{j_{2}}| \rightarrow \infty$ for each $1 \leq j_{1}, j_{2} \leq k, j_{1} \neq j_{2}$, and

and

as $n\to \infty$ where

and

Proof. The proof is similar to [34,Proposition 2.1] and [28,Lemma 1.6]; therefore, we omit it. □

Lemma 3.2. Let $X$ be a Hilbert manifold and let $F \in C^{1}(X, \mathbb{R})$ be a given functional. Let $K \subseteq X$ be compact and consider a subset.

which is invariant with respect to deformations leaving $K$ fixed. Assume that

Let $\sigma_{n} \in \mathbb{R}$ be such that $\sigma_{n} \rightarrow 0$ and $E_{n} \in \mathcal{E}$ be a sequence such that

Then, there exists a sequence $v_{n}\in X$ such that

1. $c \leq F(v_{n}) < c+\sigma_{n},$

2. $\|\nabla_{X} F(v_{n})\| < \tilde{c} \sqrt{\sigma_{n}},$

3. $\operatorname{dist}(v_{n}, E_{n}) < \tilde{c} \sqrt{\sigma_{n}},$

for some constant $\tilde{c} > 0$.

We shall prove that $I$ on $S_{c}$ possesses a kind of mountain pass geometrical structure. To this aim, we establish two preliminary lemmas.

Lemma 3.3. Assume that $h \in L^{\frac{p}{p-q}}(\mathbb{R}^{N})$ and let $u \in S_{c}$ be arbitrary but fixed. Then, we have:

(i) $I(t \star u) \rightarrow 0$ as $t \rightarrow 0$;

(ii) $I(t\star u) \rightarrow -\infty$ as $t \rightarrow +\infty$.

Proof. (i) By the Gagliardo-Nirenberg inequality (1.8), the Hölder inequality, and the assumption $(\mathbf{h}_{\mathbf{2}})$, then we have that

as $t\to 0^+$, since $p\gamma_{p}, q\gamma_{p} > 0$.

(ii) Similarly, we have that

as $t \rightarrow +\infty$, since $p \gamma_{p} > 4$. □

Again, using the Gagliardo-Nirenberg inequality and the Hölder inequality,

To understand the geometry of the functional $I$ on $S_{c}$, it is useful to consider the function $\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}$ defined by

Since $0 < q \gamma_{p} < 2 < p\gamma_{p}$, we have that $\varphi(0^{+}) = 0^{-}$ and $\varphi(+\infty) = -\infty$. The role of assumption (1.13) is clarified by the following lemma.

Lemma 3.4. Under the assumption $(\mathbf{h}_{\mathbf{2}})$, if (1.13) holds, then the function $\varphi$ has a local strict minimum at negative level and a global strict maximum at positive level. Moreover, there exists $0 < R_{1} < R_{2},$ both depending on $c$, such that $\varphi(R_{1}) = 0 = \varphi(R_{2})$ and $\varphi(t) > 0$ if and only if $t\in (R_{1}, R_{2})$.

Proof. For $t > 0$, we see that $\varphi(t) > 0$ if and only if

where

Observe that $p \gamma_{p}-q\gamma_{p} > 2-q\gamma_{p} > 0$, then $\psi$ has a unique critical point $\bar{t}$ on $(0, +\infty)$, which is a global maximum point at positive level. In fact, the expression of $\bar{t}$ is

and the maximum value of $\psi$ is

Therefore, if (1.13) holds, then $\psi(\bar{t}) > \frac{1}{q}\mathcal{C}_{N, p}^{q} c^{\frac{q(1-\gamma_{p})}{2}}\|h\| _{\frac{p}{p-q}},$ thus the equation $\varphi = 0$ has two roots $R_{1}, R_{2}$ and $\varphi$ is positive on $(R_{1}, R_{2})$. Moreover, $\varphi$ has a global maximum point $t_{2}$ at positive level. According to the expression of $\varphi$, we can deduce that $\varphi$ also has a local minimum point $t_{1}$ at negative level in $(0, R_{1})$. □

Set

By Lemmas 3.3 and 3.4, there exists a $\iota_{1} > 0$ small enough, such that

Moreover, $I^{\varphi(t_{1})} \subset\{\|\nabla u\|_{2} > R_{2}\}$ since $I(u)\geq \varphi(\|\nabla u\|_{2})$. Now we can get a mountain pass structure of $I$ on manifold $S_{c}$.

and the mountain pass value is

Remark 3.1.

where $v_{c}$ satisfies

i.e., the solution $v_{c}$ of the problem (1.6) is a mountain pass critical point of $I_{\infty}$ constrained on $S_{c}$. (see ^[35]). It is immediately seen that

Lemma 3.5. Under the assumption $(\mathbf{h}_{\mathbf{2}})$, suppose that $h$ satisfies (1.14), then there exists a $(PS)$ sequence $\{u_{n}\}$ of $\left.I\right|_{S_{c}}$, which satisfies

as $n \rightarrow \infty$, where

and

We remark that (3.13) means that there exists $\{\lambda_{n}\}_{n \geq 1}$, such that for any $\varphi \in C_{0}^{\infty}(\mathbb{R}^{N})$, there holds

Moreover, $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ and the related Lagrange multipliers $\{\lambda_{n}\}$ in (3.16) are also bounded, up to a subsequence, $\lambda_{n}\to \bar{\lambda}$, with $\bar{\lambda} > 0$.

Proof. We divide the proof into three steps.

Step 1: Existence of the Palais-Smale sequence. The existence of the $(PS)$ sequence that verifies (3.14) and (3.15) closely follows the arguments in ^[32], where the authors adapt some ideas from ^[11]. We recall the main strategy, referring to ^[32] for the details. A key tool is to set

The corresponding minimax structure of $\tilde{I}$ on $\mathbb{R} \times S_{c}$, as follows

and its minimax value is

It turns out that $\tilde{m}_{h, c} = m_{h, c}$ and that, if $(t_{n}, v_{n})_{n}$ is a $(PS)_{c}$ sequence for $\widetilde{I}$ with $t_{n}\to 0$, then $u_{n} = t_{n} \star v_{n}$ is a $(PS)_{c}$ sequence for $I$. Now, let us consider a sequence $\xi_{n} \in \Gamma$ such that

We observe that, since $I(u) = I(|u|)$ for every $u\in H^{1}(\mathbb{R}^{N})$, we can take $\xi_{n}(t) \geq 0$ in $\mathbb{R}^{N}$, for every $t \in[0, 1]$ and $n \in \mathbb{N}$. We are in a position to apply Lemma 3.2 to $\tilde{I}$ with

As a consequence, there exists a sequence $(t_{n}, v_{n}) \in \mathbb{R}\times S_{c}$ and $\tilde{c} > 0$ such that

Now, we can define

We observe that, by differentiating $\tilde{I}$ with respect to $t$, we get the "almost" Pohozaev identity (3.14), differentiating with respect to the second variable on the tangent space to $S_{c}$, and by (3.19) and $\xi_{n}(t)\ge0$ we get (3.15).

Step 2: Boundedness of the $(PS)$ sequence.

By (3.12), for the $(PS)$ sequence $\{u_{n}\}\subset S_{c}$, there holds

Combining with (3.14),

Thus $\{u_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ since $h \in L^{\frac{p}{p-q}}(\mathbb{R}^{N})$ and $\|\nabla h \cdot x\|_{\frac{2}{2-q}} < \infty$.

Step3: Positivity of the Lagrange multiplier.

By taking $u_{n}$ as a test function for (3.16), we obtain that

So

Thus the Lagrange multipliers $\{\lambda_{n}\}$ are also bounded. Next, we show that $\{\lambda_{n}\}$ has a positive lower bound. In fact, according to (3.14) and (3.16),

We also have that

Then, combined with the assumption (1.14), we have that

since

Now we prove the convergence of the $(PS)$ sequence $\{u_{n}\}$ and hence we complete the proof of Theorem 1.2.

Proof of Theorem 1.2. Next, we prove the existence of solutions of (1.1) with a positive energy level when $2+\frac{8}{N} < p < 2^{*}$. We consider the bounded $(PS)$ sequence $\{u_{n}\}$ given by Lemma 3.5. Then, there exists $u \in H^{1}(\mathbb{R}^{N})$ such that $u_{n} \rightharpoonup u$ due to the boundedness of $\{u_{n}\}$. We claim that $u_{n} \rightarrow u$ strongly in $H^{1}(\mathbb{R}^{N}).$

For any $\psi \in H^{1}(\mathbb{R}^{N})$, $\{u_{n}\}$ satisfies

Using the boundedness of $\{\lambda_{n}\}$ again, we obtain that

And hence

which implies that $\{u_{n}\}$ is a $(PS)$ sequence for $I_{\lambda}$ at level $m_{h, c}+\frac{\lambda}{2} c$, so that we can apply the Splitting Lemma 3.1, getting

Assume by contradiction that $k\geq 1$, or, equivalently, that $\|u\|_{2}^{2} < c$. In addition, if $0 < \alpha < \beta$, then $m_{\alpha} > m_{\beta}$ and $J_{\infty, 0}(\omega^{j}) \geq m_{\alpha_{j}}$ (see ^[28]). Therefore,

where $\beta: = \|u\|_{2}^{2}$, $\alpha_{j}: = \|\omega^{j}\|_{2}^{2}$. By (3.4), we have

Thus, combined with (3.25), we obtain

Since $J_{h, 0}(u), J_{\infty, 0}(\omega^{j})\geq m_{c}$, we have $m_{h, c} \geq m_{c}$, which is a contradiction of (3.11). Thus $k = 0$. That is $u_{n} \rightarrow u$ strongly in $H^{1}(\mathbb{R}^{N})$ and $u$ is a solution of Eq.(1.1). □

4.   Proof of Theorem 1.3

In this section, we assume that $2+\frac{8}{N} < p < 2^{*}$, $1\le N \le 3$, $\bar{h}(x) = -h(x)\ge 0$, and $\bar{h}(x) \not \equiv 0$. By using a min-max argument, we can find the existence of normalized solutions of Eq.(1.1). First, we show that the energy functional corresponding to Eq.(1.1) has a linking geometry. For $s \in \mathbb{R}$ and $u \in H^{1}(\mathbb{R}^{N})$, we introduce the scaling

which preserves the $L^{2}$-norm: $\|s * u\|_{2} = \|u\|_{2}$ for all $s \in \mathbb{R}$. For $\mathbb{R} > 0$ and $s_{1} < 0 < s_{2}$, which will be determined later, we set

where $B_{R} = \{x \in \mathbb{R}^{N}:|x| \leq R\}$ is the closed ball of radius $R$ around 0 in $\mathbb{R}^{N}.$ For $c > 0$, define

where $v_{c}$ satisfies

We define

To prove that the energy functional $I$ has a linking geometry, it is necessary to find the suitable $R > 0$, $s_{1} < 0 < s_{2}$ such that

at least for some suitable choice of $Q$. Now, we recall the notion of barycenter of a function $u \in H^{1}(\mathbb{R}^{N}) \backslash\{0\}$, which has been introduced in ^[36] and in ^[37]. Setting

we observe that $\nu(u)$ is bounded and continuous, so the function

is well defined, continuous, and has compact support. Therefore, we can define $\beta: H^{1}(\mathbb{R}^{N}) \backslash\{0\} \rightarrow \mathbb{R}^{N}$ as

The map $\beta$ is well defined, because $\hat{u}$ has compact support, and it is not difficult to verify that it enjoys the following properties:

(i) $\beta$ is continuous in $H^{1}(\mathbb{R}^{N}) \backslash\{0\}$;

(ii) if $u$ is a radial function, then $\beta(u) = 0$;

(iii) $\beta(tu) = \beta(u)$ for all $t \neq 0$ and for all $u \in H^{1}(\mathbb{R}^{N}) \backslash\{0\}$;

(iv) setting $u_{z}(x) = u(x-z)$ for $z \in \mathbb{R}^{N}$ and $u \in H^{1}(\mathbb{R}^{N})\backslash\{0\}$ there holds $\beta(u_{z}) = \beta(u)+z$.

Now, we define

and

It has been proved in ^[28] that

where

Lemma 4.1. $w_{c}^{r} = w_{c}^{0} = w_{c} = m_{c}$.

Proof. Clearly $\mathcal{D}_{r} \subset \mathcal{D}_{0} \subset \mathcal{D}$, so that $w_{c}^{r} \geq w_{c}^{0} \geq w_{c}$. It remains to prove that $w_{c} \geq m_{c}$ and $m_{c} \geq w_{c}^{r}$.

Arguing by contradiction, we assume that $m_{c} > w_{c}$. Then, $\max_{u \in D} I_{\infty}(u) < m_{c}$ for some $D \in \mathcal{D}$, hence $\sup_{u \in U_{\delta}(D)} I_{\infty}(u) < m_{c}$ for some $\delta > 0$, here $U_{\delta}(D)$ is the $\delta$-neighborhood of $D$. Observe that $U_{\delta}(D)$ is open and connected, so it is path-connected. Therefore, there exists a path $\sigma \in \Sigma_{c}$ such that $\max _{t \in[0, 1]} I_{\infty}(\sigma(t)) < m_{c}$, a contradiction.

The inequality $m_{c} \geq w_{c}^{r}$ follows from the fact that the set $D: = \{s\star v_{c}: s \in[s_{1}, s_{2}]\} \in \mathcal{D}_{r}$ satisfies

Lemma 4.2. $L_{c}: = \inf_{D \in \mathcal{D}_{0}} \max_{u \in D} I(u) > m_{c}$.

Proof. Using $\bar{h}(x) \geq 0$ and Lemma 4.1, we have

Now, we argue by contradiction and assume that there exists a sequence $D_{n} \in \mathcal{D}_{0}$ such that

In view of (4.1), we also have

Adapting an argument from [11,Lemma 2.4], we consider the functional

constrained to $M: = S_{c} \times \mathbb{R}$. We apply Lemma 3.2 with

and

Observe that

because $\mathcal{D} \times\{0\} \subset \mathcal{C}$, hence $w_{c} \geq \tilde{w}_{c}$, and for any $C \in \mathcal{C}$ we have $D: = \{s \star u:(u, s) \in C\} \in \mathcal{D}$ and

hence $w_{c} \leq \tilde{w}_{c}$. Hence, Lemma 3.2 yields a sequence $(u_{n}, s_{n}) \in S_{c} \times \mathbb{R}$ such that

(1) $|\tilde{I}_{\infty}(u_{n}, s_{n})-m_{c}| \rightarrow 0$ as $n \rightarrow \infty$;

(2) $\|\nabla_{S_{c} \times \mathbb{R}} \tilde{I}_{\infty}(u_{n}, s_{n})\| \rightarrow 0$ as $n \rightarrow \infty$;

(3) $\operatorname{dist}((u_{n}, s_{n}), D_{n} \times\{0\})\rightarrow 0$ as $n \rightarrow \infty.$

Then $v_{n}: = s_{n} \star u_{n} \in S_{c}$ is a $(PS)$ sequence for $I_{\infty}$ on $S_{c}$ at $m_{c}$, and there exists Lagrange multipliers $\lambda_{n} \in \mathbb{R}$ such that

as $n \rightarrow \infty$. So, combining those properties, we can infer that

and

Therefore, $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$ and $\{\lambda_{n}\}$ is bounded in $\mathbb{R}$. We may assume that $v_{n} \rightharpoonup v$ in $H^{1}(\mathbb{R}^{N})$, $\|\nabla v_{n}\|_{2}^{2} \rightarrow A^{2}$, and $\lambda_{n} \rightarrow \lambda > 0$. In fact, $\{v_{n}\}$ is a $(PS)$ sequence for $I_{\infty, \lambda}$ at level $m_{c}+\frac{\lambda}{2} c.$ As a consequence of Lemma 3.1, $v_{n}$ can be rewritten as

in $H^{1}(\mathbb{R}^{N})$, where $k \geq 0$ and $w^{j} \neq 0,$ $v$ are solutions to

and $|y_{n}^{j}| \rightarrow \infty$. Moreover, we get

and hence,

By (4.2), we have

If $v\neq 0$ and $k \geq 1$, we get $A^{2} > \|\nabla v\|_{2}^{2}$ from (4.3), we have

Similarly, we have $J_{\infty, 0}(w^{j})\ge m_{c}$. Thus,

we get a contradiction. Therefore, $k = 1$ and $v = 0$, or $k = 0$ and $v \neq 0$. If $k = 1$ and $v = 0$, then $v_{n}(\cdot+y_{n}^{1})+o(1) = w^{1}$. On the other hand, due to point (3) that $\operatorname{dist}((u_{n}, s_{n}), D_{n} \times\{0\}) \rightarrow 0$, we obtain

which contradicts the fact that $\beta$ is continuous and $|y_{n}^{1}| \rightarrow \infty$.

If $k = 0$ and $v \neq 0$, then $v_{n} \rightarrow v$ in $H^{1}(\mathbb{R}^{N})$. Using again point (3), we also have $\beta(v) = 0$. Hence, by the uniqueness, $v_{n} \rightarrow \pm v_{c}$ in $H^{1}(\mathbb{R}^{N})$. This implies

which is a contradiction. □

Lemma 4.3. For any $c > 0$, then $L_{h, c}\ge L_{c}$ holds.

Proof. Similar to [32,Proposition 3.5], so we omit it. □

Lemma 4.4. For any $c > 0$ and for any $\varepsilon > 0$, there exists $\bar{R} > 0$ and $\bar{s}_{1} < 0 < \bar{s}_{2}$ such that for $Q = B_{R} \times[s_{1}, s_{2}]$ with $R \geq \bar{R}, s_{1} \leq \bar{s}_{1}, s_{2} \geq \bar{s}_{2}$ the following holds:

Proof. We have

and

Moreover, there holds

because $\bar{h}(x)$ satisfies (1.15), thus for all $s\in \mathbb{R}$, we have

As a consequence, we deduce

provided $s_{1} < 0$ is small enough and $s_{2} > 0$ is large enough. Moreover, for $|y| = R$ large enough and $s\in [s_{1}, s_{2}]$, we choose $\alpha \in(0, 1)$ such that $\alpha (1+e^{-s_{1}}) < 1$, so that we have

The first integral is bounded by

as $R \rightarrow \infty$ and

as $R \rightarrow \infty$, which concludes the proof. □

By Lemma 4.3 and 4.4, we may choose $R > 0$ and $s_{1} < 0 < s_{2}$ such that

Therefore, $I$ has a linking geometry and there exists a $(PS)$ sequence at the level $L_{h, c}$. In order to estimate $L_{h, c}$, we have the following Lemma.

Lemma 4.5. If $|s_{1}|, s_{2}$ are large enough, then

Proof. This follows from

provided $|s_{1}|, s_{2}$ are large enough. □

By the Lemma 4.3 and Lemma 4.5, we can get

Next, we construct a bounded $(PS)$ sequence of I at $L_{h, c}$ by adopting the approach from ^[11] and Lemma 3.2. We define a auxiliary $\mathcal{C}^{1}$ functional

and

Lemma 4.6. (1) $\tilde{L}_{h, c} = L_{h, c}.$

(2) If $(u_{n}, s_{n})$ is a $(PS)$ sequence for $\tilde{I}$ at level $\tilde{L}_{h, c}$ and $s_{n} \rightarrow 0$, then $(s_{n} \star u_{n})_{n}$ is a $(PS)$ sequence for $I$ at level $L_{h, c}$.

Proof. The proof is similar to that of ^[11] and is omitted. □

Lemma 4.7. Let $\tilde{g}_{n} \in \tilde{\Gamma}_{c}$ be a sequence such that

Then, there exists a sequence $(u_{n}, s_{n}) \in S_{c} \times \mathbb{R}$ and $\tilde{c} > 0$ such that

The last inequality means:

for all

Proof. Apply Lemma 3.2 to $\tilde{I}$ with

Lemma 4.8. Under the assumption $(\mathbf{h}_{\mathbf{3}})$, then there exists a bounded $(PS)$ sequence $\{v_{n}\}$ of $\left.I\right|_{S_c}$, which satisfies

as $n \rightarrow \infty$, where

Moreover, the sequence of Lagrange multipliers satisfies, up to subsequence $\lambda_{n}\to \lambda > 0$.

Proof. First, the existence of the $(PS)$ sequence that verifies (4.6) and (4.7) closely follows the arguments in Lemma 3.5. The proof is omitted.

Next, we prove $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N}).$ By (4.4), for the $(PS)$ sequence $\{v_{n}\}\subset S_{c}$, there holds

Combining with (4.6),

Thus $\{v_{n}\}$ is bounded in $H^{1}(\mathbb{R}^{N})$, since $\|\nabla \bar{h} \cdot x\|_{\frac{2}{2-q}} < \infty$.

Then, we prove the positivity of the Lagrange multiplier in the same way as lemma 3.5. By (4.5), we obtain that

Thus, the Lagrange multipliers $\{\lambda_{n}\}$ are also bounded. In fact, according to (4.5) and (4.6), we have that

thus $\lambda > 0$ provided

So

which is given in (1.15). □

Proof of Theorem 1.3. Since $\{v_{n}\}$ is bounded, after passing to a subsequence it converges weakly in $H^{1}(\mathbb{R}^{N})$ to $v\in H^{1}(\mathbb{R}^{N})$. By (4.7) and weak convergence, $v$ is a nonnegative weak solution of

such that $\beta: = \|v\|_{2}^{2} \leq c$, where $A^{2}: = \lim_{n \rightarrow \infty}\|\nabla v_{n}\|_{2}^{2}$. We note that $\{v_{n}\}$ is a bounded $(PS)$ sequence of $I_{\lambda}$ at level $L_{h, c}+\frac{\lambda}{2}c$, therefore, by Lemma 3.1, there exists an integer $k\geq 0$, $k$ non-trivial solutions $w^{1}, w^{2}, \ldots, w^{k}$ to the equation

and $k$ sequences $\{y_{n}^{j}\} \in H^{1}(\mathbb{R}^{N}), 1 \leq j \leq k$, such that $|y_{n}^{j}| \rightarrow \infty$ as $n \rightarrow \infty$.

Moreover, we have

and

as $n \rightarrow \infty$. It remains to show $k = 0$, so that $v_{n} \rightarrow v$ strongly in $H^{1}(\mathbb{R}^{N})$ and we are done. Thus, by contradiction, we can assume that $k\geq 1$, or equivalently $\beta < c$.

First, we exclude the case $v = 0$. In fact, if $v = 0$ and $k = 1$, we have $w^{1} > 0$ and $\|w^{1}\|_{2}^{2} = c$ and $\|\nabla w^{1}\|_{2}^{2} = A^{2}$ so that (4.13) would give $L_{h, c} = m_{c}$, which is not possible due to Lemma 5.3. On the other hand, if $k \geq 2$, we get $J_{\infty, 0}(w^{j}) \geq m_{\alpha_{j}}\left(\alpha_{j}: = \|w^{j}\|_{2}^{2}\right)$ and $m_{\alpha_{j}} > m_{c}$, thus $L_{h, c} > 2m_{c}$, which contradicts with Lemma 4.5.

Therefore, from now on, we will assume $v\neq0$ and $k\ge 1$. From (4.13) and $I(v_{n})\to L_{h, c}$, we deduce

Using (4.12), we have

Then, from $A^{2} > \|\nabla v\|_{2}^{2}$ and $\bar{h}(x) \ge 0$, we have

Similarly, we have $J_{\infty, 0}(w^{j})\ge m_{c}$. Thus,

we get a contradiction. Thus $k = 0$ and $\{v_{n}\}$ converges strongly to $v$ in $H^{1}(\mathbb{R}^{N})$. □

Author contributions

Xin Qiu: Writing-original draft, Writing-review & editing; Zeng Qi Ou: Supervision, Formal Analysis; Ying Lv: Writing-review & editing, Methodology, Supervision.

Use of AI tools declaration

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

Acknowledgments

This research is supported by National Natural Science Foundation of China (No.12371120), Innovation Research 2035 Pilot Plan of Southwest University(SWU-XDPY22015) and Chongqing Postgraduate Research and Innovation Programme(CYS240130).

Conflict of interest

The authors declare there is no conflict of interest.