Existence and Multiplicity of Normalized Solutions with Positive Energy for the Kirchhoff Equation

Abstract

In this paper, we investigate the existence and multiplicity of normalized solutions for the following Kirchhoff equation,

$$\begin{aligned} \left\{ \begin{array}{l} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u), \quad \text{ in } \mathbb {R}^{3}, \\ \int _{\mathbb {R}^{3}}|u|^{2} d x=c, \end{array}\right. \end{aligned}$$

(P)

where a, b, c are positive constants and $\lambda \in \mathbb {R}$ is an unknown parameter that appears as a Lagrange multiplier. Two normal solutions, manifesting as a local minimizer or mountain pass solution, have been obtained under the mass subcritical conditions on the nonlinearity f and some suitable mass c. Additionally, we employ the Symmetric Mountain Pass Theorem to establish the multiplicity of normalized solutions for problem (P). To the best of our knowledge, we extend and complement the research success in the Kirchhoff equation for a general nonlinearity with weaker subcritical mass growth.

1 Introduction

This paper investigates a Kirchhoff type problem with a general nonlinearity,

$$\begin{aligned} -\left( a+b \int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u-\lambda u=f(u), \quad \text {in } \mathbb {R}^3, \end{aligned}$$

(1.1)

where a, b are positive constants and $\lambda \in \mathbb {R}$ is an parameter. The Eq. (1.1) is closely related to the equation

$$\begin{aligned} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2} d x\right) \Delta u=f(x, u), \end{aligned}$$

(1.2)

which is the stationary analog of the equation

$$\begin{aligned} u_{t t}-\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2} d x\right) \Delta u=f(x, u). \end{aligned}$$

(1.3)

The Eq. (1.3) was initially proposed by Kirchhoff in 1883 as an extension of the classical D’Alembert wave equation. Subsequent to the pioneering research by Lions [1], the Kirchhoff type problem gained substantial attention from the research community. Researchers then devoted their efforts to studying its steady-state model, leading to significant research progress as documented in [2,3,4,5,6].

Currently, two main perspectives exist regarding the parameter $\lambda$ in Eq. (1.1). The first perspective treats $\lambda$ as a constant. In this case, solutions of Eq. (1.1) correspond to critical points of the associated action functional in the function space. This approach has been extensively studied, as documented in [4, 7,8,9,10,11,12], along with other relevant references.

The second viewpoint regards the frequency $\lambda$ as an unknown quantity in Eq. (1.1). Physicists are particularly interested in solutions that satisfy the normalized condition: $\int _{\mathbb {R}^{3}}|u|^{2} d x=c$, for a priori given c, since the mass admits a clear physical meaning. By employing critical point theory to examine solutions of Eq. (1.1) that satisfying the normalized condition. It is sufficient to analyze the critical points of the following $C^1$ functional,

$$\begin{aligned} I(u)=\frac{a}{2} \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx+\frac{b}{4}\left( \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) ^{2}-\int _{\mathbb {R}^{3}} F(u)dx, \end{aligned}$$

(1.4)

constrained on the following $L^{2}$ spheres in $H^{1}\left( \mathbb {R}^{3}\right)$,

$$\begin{aligned} S_{c}:=\left\{ u \in H^{1}\left( \mathbb {R}^{3}\right) \mid \int _{\mathbb {R}^{3}}|u|^{2}dx=c\right\} . \end{aligned}$$

To accommodate the constraint $S_{c}$, it becomes essential to define the dilation, expressed as

$$\begin{aligned} (s \star u)(x):=e^{\frac{3}{2} s} u\left( e^{s} x\right) , \quad \text{ for } s\in \mathbb R\ \ \text{ and } x \in \mathbb {R}^{3}. \end{aligned}$$

It’s worth noting that the element $s \star u \in H^{1}\left( \mathbb {R}^{3}\right)$ preserves the $L^{2}$ norm for $s \in \mathbb {R}$, and the mapping that sends $(s, u) \in \mathbb {R} \times H^{1}\left( \mathbb {R}^{3}\right)$ to $s \star u$ is continuous.

For any fixed $c>0$, we call $\left( u_{c}, \lambda _{c}\right) \in H^{1}\left( \mathbb {R}^{3}\right) \times \mathbb {R}$ a couple of solution to problem (P) if $u_{c}$ is a critical point of $I_{|{S_{c}}}$ and $\lambda _{c}$ is the associated Lagrange multiplier. It is evident that solutions to problem (P) can be derived as critical points of the functional I under the constraint of $S_{c}$. So it is nature to consider the following minimization problem,

$$\begin{aligned} I_{c}:=\inf _{u \in S_{c}} I(u). \end{aligned}$$

(1.5)

Moreover, we define $v \in S_{c}$ as an energy ground state solution for problem (P) if it satisfies the following conditions,

$$\begin{aligned} \left( I_{\mid S_{c}}\right) ^{\prime }(v)=0 \quad \text{ and } \quad I(v)=\inf \left\{ I(u) \mid \left( I_{\mid S_{c}}\right) ^{\prime }(u)=0\right\} . \end{aligned}$$

When the nonlinearity is a pure power term i.e. $f(u)=|u|^{p-2}u$, Ye [13] studied the following minimization problem (1.5), for any $c>0$ and $N\le 3$. The author demonstrated that the mass critical exponent for this minimization problem is $2+\frac{8}{N}$ by using the well-known Gagliardo Nirenberg inequality and mass preserving scaling arguments. In other words, $I_{c}>-\infty$ for $p\in (2, 2+\frac{8}{N})$ and $I_{c}=-\infty$ for $p\in (2+\frac{8}{N}, \frac{2N}{N-2})$. Recently, Qi and Zou [14] focused on determining the exact number of positive normalized solutions for $p\in (2,\frac{N-2}{2N})$, $N\ge 3$, and observed novel phenomena arising from the nonlocal nature of the equation. In addition to the normalized solutions previously discussed, further results related to $p \in \left[ 2+\frac{8}{N}, \frac{2N}{N-2}\right)$ are presented in [9, 13, 15, 16].

For general nonlinearity f with a mass supercritical growth, Xie and Chen [17] demonstrated the existence and multiplicity of solutions for problem (P). In [18], He et al. established the existence of ground state normalized solutions for problem (P) with a potential. The normalized solutions were achieved through the investigation of a constraint problem on a Nehari-Pohozaev manifold. In [19], Zeng et al. determined the existence, non-existence and multiplicity of positive normalized solutions for problem (P) in different dimensional cases. In the context of general mass subcritical nonlinearity, Ye and Zhang [20] established the precise existence and nonexistence of global constraint minimizers, by the concentration compactness principle. In [21], Chen et al. confirmed the existence of normalized solutions for nonautonomous Kirchhoff equation.

To provide clarity in our subcritical mass setting and for the sake of future discussions, we make the assumption that the nonlinear term $f \in C(\mathbb {R}, \mathbb {R})$ satisfies the following conditions:

$\left( f_{0}\right)$:

 There exists $C>0$, such that $|f(t)| \le C\left( |t|+|t|^{5}\right)$, for $t \in \mathbb {R}$.

$\left( f_{1}\right)$:

 $\lim _{t \rightarrow 0} \frac{f(t)}{t}=0$.

$\left( f_{2}\right)$:

 $\lim \sup _{|t| \rightarrow \infty } \frac{f(t) t}{|t|^{\frac{14}{3}}} \le 0; \quad \left( f_{2}\right) ^{\prime } ~ \lim _{|t| \rightarrow \infty } \frac{f(t) t}{|t|^{\frac{14}{3}}}=0$.

$\left( f_{3}\right)$:

 There exists $t_{0}>0$ such that $F\left( t_{0}\right) >0$, where $F(t)=\int _{0}^{t}f(s)ds$ for $t\in \mathbb {R}$.

$\left( f_{4}\right)$:

 $\lim \sup _{t \rightarrow 0} \frac{F(t)}{|t|^{\frac{10}{3}}} \le 0;~ \left( f_{4}\right) ^{\prime } ~ \lim _{t \rightarrow 0} \frac{F(t)}{|t|^{\frac{10}{3}}}=0$.

$(f_5)$:

 There exists $\vartheta \in \left( 2,6\right)$ such that $\vartheta F(t) \ge f(t) t$, for any $t \in \mathbb {R}$.

Then we have the first result about the global minimization problem as follows.

Theorem 1.1

Assume that f satisfies $(f_{0})-(f_{3})$. There exists $c^{*} \ge 0$ such that $I_{c}=0$ for all $0<c \le c^{*}$ and $I_{c}<0$ for all $c>c^{*}$. Then the map $c \mapsto I_{c}$ is continuous and non-increasing. Moreover,

$\mathrm{(i)}$:

 $I_{c}$ has a minimizer for each $c>c^{*}$.

$\mathrm{(ii)}$:

 If $(f_{4})$ holds, then $c^{*}>0$, $I_{c}$ has no minimizer for each $0<c<c^{*}$ and $I_{c^{*}}$ has a minimizer. In particular, the minimizer is an energy ground state solution of problem (P), with the associated Lagrange multiplier being negative.

Remark 1.1

It should be mentioned that the authors obtained a similar result with assumptions of $(f_{1})$, $(f_{2})^{\prime }$ and $(f_{3})$ in Theorem 1.1 of [20]. Moreover, $c^*>0$ by $(f_{4})^{\prime }$ their Theorem 1.2(3). Obviously, $(f_{2})$ and $(f_{4})$ are weaker than $(f_{2})^{\prime }$ and $(f_{4})^{\prime }$ respectively. From this aside, their results has been improved by our Theorem 1.1.

Remark 1.2

Recently, the following Kirchhoff problem has received attention

$$\begin{aligned} \left\{ \begin{array}{l} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2} d x\right) \Delta u=\lambda u+\mu |u|^{q-2} u+|u|^{p-2} u, \quad \text{ in } \mathbb {R}^{3}, \\ \int _{\mathbb {R}^{3}}|u|^{2} d x=c, \end{array}\right. \end{aligned}$$

(P1)

where a, b, $c>0$, $2<q$, $p\le 6$. For the case that $\mu <0$, some of results on normalized solutions to this problem can been found in [22,23,24]. Specifically, we consider the above nonlinearity $f_{1}(t):= |t|^{p-2} t + \mu |t|^{q-2} t$ with $\mu < 0$, and $\frac{10}{3}< p<\frac{14}{3}< q \le 6$. It is evident to check that this function satisfies conditions $(f_{0})-(f_{5})$, but not $(f_{2})^{\prime }$. To the best of our knowledge, there is few results established for problem (P1) with $f_{1}(u)$ and our paper addresses this issue, as demonstrated in Theorems 1.1–1.4.

Theorem 1.2

Assume that f satisfies $(f_{0})-(f_{4})$. For any $c \in \left( 0, c^{*}\right]$, we define $\bar{I}_{c}:=\inf _{u \in S_{c}^{\rho }} I(u)$, where

$$\begin{aligned} S_{c}^{\rho }:=\left\{ u \in S_{c} \mid \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx>\rho \left( c^{*}\right) \right\} . \end{aligned}$$

Then the map $c \mapsto \bar{I}_{c}$ is continuous and non-increasing. Moreover, there exists $c^{* *} \in \left( 0, c^{*}\right)$ such that when $c\in \left( c^{* *}, c^{*}\right]$, the local infimum $\bar{I}_{c}$ is achieved by some $v\in S_{c}^{\rho }$, with

$$\begin{aligned} I(v)=\left\{ \begin{array}{ll} \bar{I}_{c}>0 &{} \text{ for } c \in \left( c^{* *}, c^{*}\right) , \\ \bar{I}_{c}=\inf _{u \in S_{c}} I(u)=0 &{} \text{ for } c=c^{*}. \end{array}\right. \end{aligned}$$

If in addition

$$\begin{aligned} \limsup _{t \rightarrow 0} \frac{f(t) t-2 F(t)}{|t|^{\frac{10}{3}}} \le 0, \end{aligned}$$

(1.6)

then $v \in S_{c}^{\rho }$ is an energy ground state solution of problem (P).

When assuming condition $(f_5)$ in addition, our following result demonstrates the existence of a positive energy solution of saddle type for any $c > c^{**}$. To avoid any misinterpretation when analyzing Theorem 1.3, it is essential to highlight that conditions $(f_4)$ and $(f_5)$ imply (1.6).

Theorem 1.3

Assume that f satisfies $(f_{0})-(f_{5})$. For any $c>c^{* *}$, problem (P) admits a radial solution $w \in S_{c}$ which corresponds to a mountain pass level and satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} I(w)>0>I(v) &{} \text{ for } c>c^{*}, \\ I(w)>0=I(v) &{} \text{ for } c=c^{*}, \\ I(w)>I(v)>0 &{} \text{ for } c \in \left( c^{* *}, c^{*}\right) . \end{array}\right. \end{aligned}$$

Here, $v \in S_{c}$ is the ground state solution of problem (P) given by Theorem 1.1 for $c\ge c^{*}$ and Theorem 1.2 for $c \in \left( c^{* *}, c^{*}\right]$, respectively.

Jeanjean and Lu [25] utilized a new version of the Symmetric Mountain Pass Theorem to demonstrate the existence of increasing numbers of normalized radial solutions with positive energies as the mass increases. Under the framework of Theorem 1.3, we establish the existence of a finite number of positive energy solutions for mass subcritical problems when f is an odd function.

Theorem 1.4

Assume that f is an odd function satisfying conditions $(f_{0}) -(f_{5})$. For each $k \in \mathbb {N}^{+}$, there exists $c_{k} > 0$ such that when $c > c_{k}$, problem (P) has at least k distinct radial solutions with positive energies.

The structure of this paper is organized as follows. In Sect. 2, we present some preliminary lemmas. Sections 3, 4 and 5 are dedicated to proving Theorems 1.1–1.4, respectively.

Notation. Throughout this paper, we denote C, $C_{i}$ for various positive constants, whose specific values may vary from line to line but are not critical to the problem analysis. The norm $\Vert \cdot \Vert _{s}$ represents the standard norm in $L^{s}(\mathbb {R}^{3})$ for $s \in [2, \infty ]$. $H_{r}^{1}(\mathbb {R}^{3})$ denotes the space of radially symmetric functions in $H^{1}(\mathbb {R}^{3})$. $S_{c,r}:= S_{c} \cap H_{r}^{1}(\mathbb {R}^{3})$. $H^{-1}(\mathbb {R}^{3})$ is the dual space of $H^{1}(\mathbb {R}^{3})$. $B_{r}(u)$ is an open ball centered at u with a radius of $r>0$. We use $o_{n}(1)$ to represent a quantity that approaches 0 as $n \rightarrow \infty$.

2 Preliminary Results

In preparation for the proofs of our main theorems, this section introduces several technical results. We need the following Gagliardo Nirenberg inequality, and the proof can be found in [26, 27].

Lemma 2.1

Let $p \in (2,6)$. Then there exists a constant $C_{p}$ such that

$$\begin{aligned} \Vert u\Vert _{p} \le C_{p}\Vert \nabla u\Vert _{2}^{\gamma _{p}}\Vert u\Vert _{2}^{\left( 1-\gamma _{p}\right) }, \quad \text {for any} \quad u \in H^{1}\left( \mathbb {R}^{3}\right) \end{aligned}$$

where $\gamma _{p}=\frac{3(p-2)}{2 p}$.

Lemma 2.2

If $u \in S_{c}$ be a solution to problem (P), then $P(u)=0$, where

$$\begin{aligned} P(u):=a\Vert \nabla u\Vert _{2}^{2}+b\Vert \nabla u\Vert _{2}^{4}-\frac{3}{2} \int _{\mathbb {R}^{3}}[f(u) u-2 F(u)]d x. \end{aligned}$$

Proof

Let u be a solution to problem (P), we firstly have that

$$\begin{aligned} a\Vert \nabla u\Vert _{2}^{2}+b\Vert \nabla u\Vert _{2}^{4}-\lambda \Vert u\Vert _{2}^{2}=\int _{\mathbb {R}^{3}} f(u) u d x. \end{aligned}$$

(2.1)

Secondly, u satisfies the Pohozaev identity (see Lemma 2.1 in [28])

$$\begin{aligned} Q(u)=a\Vert \nabla u\Vert _{2}^{2}+b\Vert \nabla u\Vert _{2}^{4}-3 \lambda \Vert u\Vert _{2}^{2}-6\int _{\mathbb {R}^{3}} F(u) d x=0. \end{aligned}$$

(2.2)

Eliminate the unknown parameter $\lambda$, we obtain that $P(u)=0$. $\square$

Lemma 2.3

Assume that f satisfies $(f_{0})-(f_{3})$. Then the following statements hold:

$\mathrm{(i)}$:

 For any bounded sequence $\left\{ u_{n}\right\}$ in $H^{1}\left( \mathbb {R}^{3}\right)$,

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) d x=0,~ \text {if} ~\lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}=0, \end{aligned}$$

and

$$\begin{aligned} \limsup _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) d x \le 0,~ \text {if} ~ \lim _{n \rightarrow \infty }\left\| u_{n}\right\| _{\frac{14}{3}}=0. \end{aligned}$$

$\mathrm{(ii)}$:

  I is coercive on $S_{c}$.

$\mathrm{(iii)}$:

 For any $c>0$, there exists $\rho (c)>0$ small enough such that, for all $u \in H^{1}\left( \mathbb {R}^{3}\right)$ satisfying both $\Vert u\Vert _{2}^{2} \le c$ and $\Vert \nabla u\Vert _{2}^{2} \le 4\rho (c)$, we have

$$\begin{aligned} I(u) \ge \frac{a}{4}\Vert \nabla u\Vert _{2}^{2}, \quad \text{ if } (f_4) \text{ is } \text{ satisfied }, \end{aligned}$$

and

$$\begin{aligned} P(u) \ge \frac{a}{2}\Vert \nabla u\Vert _{2}^{2}, \quad \text{ when } 1.6 \text{ holds }. \end{aligned}$$

Proof

The proofs of (i) and (ii) are standard. Next, we prove (iii). By $\left( f_{2}\right)$, $\left( f_{4}\right)$, for any $\varepsilon >0$ there exists $R_{\varepsilon }>0$ such that

$$\begin{aligned} F(t) \le \varepsilon |t|^{\frac{10}{3}} \text{ for } |t| \le R_{\varepsilon }^{-1} \quad \text{ and } \quad F(t) \le \varepsilon |t|^{\frac{14}{3}} \text{ for } |t| \ge R_{\varepsilon }. \end{aligned}$$

(2.3)

By $\left( f_{0}\right)$, (2.3), we deduce that, there exists $C_{\varepsilon }>0$ such that

$$\begin{aligned} F(t) \le \varepsilon |t|^{\frac{10}{3}}+C_{\varepsilon }|t|^{\frac{14}{3}} \text{ for } t \in \mathbb {R}. \end{aligned}$$

(2.4)

For any $\Vert u\Vert _{2}^{2} \le c$, invoking the Gagliardo Nirenberg inequality, we deduce that

$$\begin{aligned} \begin{aligned} I(u) =&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4} -\int _{\mathbb {R}^{3}} F(u) d x \\ \ge&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4} - \epsilon \Vert u\Vert _{\frac{10}{3}}^{\frac{10}{3}} -C_{\epsilon }\Vert u\Vert _{\frac{14}{3}}^{\frac{14}{3}} \\ \ge&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4}-\epsilon C \Vert \nabla u\Vert _{2}^{2}-C_{\epsilon }C \Vert \nabla u\Vert _{2}^{4}\\ =&\left( \frac{a}{2}-\epsilon C\right) \Vert \nabla u\Vert _{2}^{2}+\left( \frac{b}{4}-C_{\epsilon }C\right) \Vert \nabla u\Vert _{2}^{4}\ge \frac{a}{4}\Vert \nabla u\Vert _{2}^{2}, \end{aligned} \end{aligned}$$

where $\Vert \nabla u\Vert _{2}^{2} \le 4\rho (c)$ and $\epsilon$, $\rho (c)$ are sufficiently small. Similarly, $P(u) \ge \frac{a}{2}\Vert \nabla u\Vert _{2}^{2}$ also holds. $\square$

The following lemmas in our study present a deformation result that enables us to employ genus theory and an appropriate version of the Symmetric Mountain Pass theorem from [29] to establish Theorem 1.4. To introduce this lemma, it is crucial to revisit the Palais-Smale-Pohozaev condition at a level $m \in \mathbb {R}$, also referred to as the $(PSP)_{m}$ condition, which was initially introduced in [30, 31].

Definition 2.1

Given an arbitrary $m\in \mathbb {R}$, we say that the constrained functional $I_{\mid S_{c,r}}$ satisfies the $(\textrm{PSP})_{m}$ condition if any sequence $\{u_{n}\} \subset S_{c,r}$ that fulfills the conditions:

$$\begin{aligned} I\left( u_{n}\right) \rightarrow m, \quad \left( I_{\mid S_{c,r}}\right) ^{\prime }(u_{n}) \rightarrow 0 \quad \text{ and } \quad P(u_{n}) \rightarrow 0 \end{aligned}$$

(2.5)

leads to the existence of a strongly convergent subsequence in $H_{r}^{1}\left( \mathbb {R}^{3}\right)$. For convenience, we shall henceforth refer to any sequence $\left\{ u_{n}\right\}$ that satisfies (2.5) as a $(PSP)_{m}$ sequence of $I_{\mid S_{c,r}}$.

For a given $m \in \mathbb {R}$, we define

$$\begin{aligned} I^{m}:= \left\{ u \in S_{c,r} \mid I(u) \le m\right\} , \end{aligned}$$

and denote by $K^{m}$ the set of critical points of $I_{\mid S_{c,r}}$ at level m. The subsequent lemma was established by Ikoma and Tanaka. Detailed proofs can be found in [25, 31].

Lemma 2.4

Assume that f satisfies $(f_{0})-(f_{2})$. If the constrained functional $I_{\mid S_{c,r}}$ satisfies the $(\textrm{PSP})_{m}$ condition at some level $m \in \mathbb {R}$, then for any neighborhood $\mathcal {O} \subset S_{c,r}$ of $K^{m}(\mathcal {O}=\emptyset$ if $K^{m}=\emptyset )$ and any $\bar{\varepsilon }>0$, there exists $\varepsilon \in (0, \bar{\varepsilon })$ and $\eta \in C([0,1] \times S_{c,r}, S_{c,r})$ such that the following properties hold:

$\mathrm{(i)}$:

 $\eta (0, u)=u$ for any $u \in S_{c,r}$.

$\mathrm{(ii)}$:

 $\eta (t, u)=u$ for any $t \in [0,1]$ if $u \in I^{m-\bar{\varepsilon }}$.

$\mathrm{(iii)}$:

 $t \mapsto I(\eta (t, u))$ is non-increasing for any $u \in S_{c,r}$.

$\mathrm{(iv)}$:

 $\eta \left( 1, I^{m+\varepsilon } \backslash \mathcal {O}\right) \subset I^{m-\varepsilon }$ and $\eta \left( 1, I^{m+\varepsilon }\right) \subset I^{m-\varepsilon } \cup \mathcal {O}$.

$\mathrm{(v)}$:

 $\eta (t,-u)=-\eta (t, u)$ for any $(t, u) \in [0,1] \times S_{c,r}$ when f is odd.

To make use of the deformation lemma described above in our upcoming proofs, it is crucial to establish the $(\textrm{PSP})_{m}$ condition at a certain level $m \in \mathbb {R}$.

Lemma 2.5

Assume that f satisfies $(f_0)-(f_3)$ and $(f_5)$, the constrained functional $I_{\mid S_{c,r}}$ satisfies the $(\textrm{PSP})_{m}$ condition for any $m \ne 0$.

Proof

Let $\left\{ u_{n}\right\} \subset S_{c,r}$ be an arbitrary $(\textrm{PSP})_{m}$ sequence with $m\ne 0$ and the sequence $\left\{ u_{n}\right\}$ is bounded in $H_{r}^{1}\left( \mathbb {R}^{3}\right)$, since functional I is coercive by Lemma 2.3 (ii). Then there exists $u \in H_{r}^{1}\left( \mathbb {R}^{3}\right)$ such that, up to a subsequence,

$$\begin{aligned}{} & {} u_{n} \rightharpoonup u \text{ in } H_{r}^{1}\left( \mathbb {R}^{3}\right) ,\quad u_{n} \rightarrow u \text{ in } L^{p}\left( \mathbb {R}^{3}\right) , p \in (2,6),\nonumber \\{} & {} \quad u_{n}(x) \rightarrow u(x) \text{ a.e. } x\in \mathbb {R}^{3}. \end{aligned}$$

(2.6)

Moreover, passing to a subsequence, we may assume that

$$\begin{aligned} A:=\lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}^{2},\quad \lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) d x\quad \text{ and } \lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} f\left( u_{n}\right) u_{n} d x~ \text {exist}. \end{aligned}$$

By a standard argument, see Lemma 3 in [32], as $n \rightarrow \infty$, we know

$$\begin{aligned} I^{\prime }_{\mid S_{c,r}}\left( u_{n}\right) \rightarrow 0 \text{ in } H_{r}^{-1}\left( \mathbb {R}^{3}\right) \Longleftrightarrow I^{\prime }\left( u_{n}\right) -\left\langle I^{\prime }\left( u_{n}\right) , u_{n}\right\rangle u_{n} \rightarrow 0 \text{ in } H_{r}^{-1}\left( \mathbb {R}^{3}\right) . \end{aligned}$$

Then, for any $\omega \in H_{r}^{1}\left( \mathbb {R}^{3}\right)$, we have

$$\begin{aligned}&\left\langle I^{\prime }\left( u_{n}\right) -\left\langle I^{\prime }\left( u_{n}\right) , u_{n}\right\rangle u_{n}, \omega \right\rangle =\left( a+b \Vert \nabla u_{n}\Vert _{2}^{2}\right) \int _{\mathbb {R}^{3}} \nabla u_{n} \nabla \omega dx\nonumber \\&\quad -\int _{\mathbb {R}^{3}}f(u_{n}) \omega dx-\lambda _{n} \int _{\mathbb {R}^{3}} u_{n} \omega dx, \end{aligned}$$

(2.7)

where

$$\begin{aligned} \lambda _{n}=\frac{1}{\left\| u_{n}\right\| _{2}^{2}}\left[ \left( a+b \Vert \nabla u_{n}\Vert ^{2}\right) \Vert \nabla u_{n}\Vert _{2}^{2}-\int _{\mathbb {R}^{3}}f(u_{n})u_{n}dx\right] . \end{aligned}$$

(2.8)

Since each term in the right hand of (2.8) is bounded, there exists $\lambda \in \mathbb {R}$ such that $\lambda _{n}\rightarrow \lambda$ as $n\rightarrow \infty$.

In order to prove that $\lambda <0$, for the number $\vartheta \in \left( 2,6\right)$ given in $(f_5)$, we set

$$\begin{aligned} \beta :=\frac{6-\vartheta }{3(\vartheta -2)}>0. \end{aligned}$$

From (2.8), the fact that $P(u_{n}) \rightarrow 0$ and $(f_5)$, it follows

$$\begin{aligned} \begin{aligned} \lambda c^{2}&=\lambda _{n} c^{2}+o_{n}(1)\\&=-\beta (a\Vert \nabla u_{n} \Vert _{2}^{2} + b\Vert \nabla u_{n} \Vert _{2}^{4})+(\beta +1)(a\Vert \nabla u_{n} \Vert _{2}^{2} + b\Vert \nabla u_{n} \Vert _{2}^{4})-\int _{\mathbb {R}^{3}} f(u_{n})u_{n}dx\\&=-\beta (a\Vert \nabla u_{n} \Vert _{2}^{2} + b\Vert \nabla u_{n} \Vert _{2}^{4})+(\beta +1)P(u_{n})+\frac{2}{\vartheta -2}\left( \int _{\mathbb {R}^{3}}(f(u_{n})u_{n}-\vartheta F(u_{n}))dx\right) \\&\le -\beta (a\Vert \nabla u_{n} \Vert _{2}^{2} + b\Vert \nabla u_{n} \Vert _{2}^{4}). \end{aligned} \end{aligned}$$

We now claim that

$$\begin{aligned} \lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}>0. \end{aligned}$$

(2.9)

If not, then $\lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) d x=0$ by Lemma 2.3 (i) and we obtain thus a contradiction

$$\begin{aligned} 0 \ne m=\lim _{n \rightarrow \infty } I\left( u_{n}\right) =0. \end{aligned}$$

In view of (2.9), for $n\rightarrow \infty$, we deduce that $\lambda <0$ and

$$\begin{aligned} -(a+b A) \Delta u-\lambda u-f\left( u\right) = 0, \quad \text {in}~ H^{-1}\left( \mathbb {R}^{3}\right) . \end{aligned}$$

(2.10)

By standard steps, we obtain following equality and details in Lemma 2.5 in [25],

$$\begin{aligned} \int _{\mathbb {R}^{3}} f(u) u d x \ge \lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} f\left( u_{n}\right) u_{n} d x. \end{aligned}$$

(2.11)

Thus, by (2.8-(2.11)) and the fact that $\lambda _{n} \rightarrow \lambda <0$, we obtain

$$\begin{aligned} \begin{aligned} (a+b A) \Vert \nabla u\Vert _{2}^{2}-\lambda \Vert u\Vert _{2}^{2}&=\int _{\mathbb {R}^{3}} f(u) u d x \ge \lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} f\left( u_{n}\right) u_{n} d x \\&=(a+b A) \lim _{n \rightarrow \infty } \Vert \nabla u_{n}\Vert _{2}^{2}-\lambda c. \end{aligned} \end{aligned}$$

We conclude that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert \nabla u_{n}\Vert _{2}^{2} =\Vert \nabla u\Vert _{2}^{2}, \quad c=\Vert u\Vert _{2}^{2}=\lim _{n \rightarrow \infty } \Vert u_{n}\Vert _{2}^{2}, \end{aligned}$$

which implies $u_{n} \rightarrow u$ in $H_{r}^{1}\left( \mathbb {R}^{3}\right)$. The proof is complete. $\square$

3 The Proof of Theorem 1.1

By Lemma 2.3 (ii), $I_{c}=\inf _{u \in S_{c}} I(u)$ is well defined.

Lemma 3.1

Assume that f satisfies $(f_0)-(f_3)$. Then for any $c>0$, $I_{c} \le 0$.

Proof

For any $c>0$ and $u \in S_{c}$, define $u_{t}(x):=t^{\frac{3}{2}} u(t x)$ for any $t>0$. Subsequently, $u_{t} \in S_{c}$ and by $(f_{1})$, we get

$$\begin{aligned} I_{c} \le I\left( u_{t}\right) =t^{2} \frac{a}{2} \Vert \nabla u\Vert _{2}^{2}+t^{4} \frac{b}{4}\Vert \nabla u\Vert _{2}^{4}-\int _{\mathbb {R}^{3}} \frac{F\left( t^{\frac{3}{2}} u\right) }{\left| t^{\frac{3}{2}} u\right| ^{2}}|u|^{2}dx \rightarrow 0, \end{aligned}$$

as $t \rightarrow 0^{+}$. Hence $I_{c} \le 0$ for all $c>0$. $\square$

For the sake of brevity in exposition, the proof of the following lemma can be referred to Lemma 4.1 in Sect. 4.

Lemma 3.2

Assume that f satisfies $(f_0)-(f_3)$. Then the following conclusions are true.

$\mathrm{(i)}$:

 The function $c \mapsto I_{c}$ is continuous on $\left( 0, +\infty \right)$;

$\mathrm{(ii)}$:

 The function $c \mapsto I_{c}$ is non-increasing on $\left( 0, +\infty \right)$;

Lemma 3.3

Assume that f satisfies $(f_0)-(f_3)$. Then there exists $c^{*} \ge 0$ such that $I_{c}<0$ if $c>c^{*}$. Moreover, if $c^{*}>0$, then $I_{c}=0$ for all $0<c \le c^{*}$.

Proof

Given any $u \in S_{1}$, let $u_{c}(x):=u(c^{-\frac{1}{3}} x)$, for all $c>0$. Then $u_{c} \in S_{c}$, we have

$$\begin{aligned} I_{c} \le I\left( u_{c}\right) =c^{\frac{1}{3}} \frac{a}{2} \Vert \nabla u\Vert _{2}^{2}+c^{\frac{2}{3}} \frac{b}{4}\Vert \nabla u\Vert _{2}^{4}-c \int _{\mathbb {R}^{3}} F(u)dx. \end{aligned}$$

As $c \rightarrow +\infty$, $I_{c} \le I\left( u_{c}\right) \rightarrow -\infty$. Consequently, $I_{c}<0$ for sufficiently large values of c. Therefore, the set of c such that $I_{c}<0$ in $(0,+\infty )$ is not empty.

Define

$$\begin{aligned} c^{*}:=\inf \left\{ c>0 \mid I_{c}<0\right\} . \end{aligned}$$

Consequently, $c^{*}\ge 0$ is well defined and $I_{c}<0$ if $c>c^{*}$. If $c^{*}>0$, we can conclude from Lemma 3.2 that $I_{c}=0$ for all $c<c^{*}$ and $I_{c^{*}}=0$. $\square$

The proof of following lemma is similar to Lemma 2.5 in [20] and we omit the proof here.

Lemma 3.4

Assume that f satisfies $(f_0)-(f_3)$. For each $c>c^{*}$, it holds $I_{c}<I_{\alpha }+I_{c-\alpha }$ for any $0<\alpha <c$.

Proof of Theorem 1.1 (i) When $c>c^{*}$, it follows from Lemma 3.3 that $I_{c}<0$. Considering a sequence $\left\{ u_{n}\right\} \subset S_{c}$ be a minimizing sequence of $I_{c}$, then by Lemma 2.3 (ii), $\left\{ u_{n}\right\}$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$. Let $\delta :=\lim _{n \rightarrow \infty } \sup _{y \in \mathbb {R}^{3}} \int _{B_{1}(y)}\left| u_{n}\right| ^{2}dx$ and then $\delta \ge 0$. In the case $\delta =0$, the vanishing lemma (see Lemma I.1 in [33]) implies that, $u_{n} \rightarrow 0$ in $L^{p}\left( \mathbb {R}^{3}\right)$, for $2<p<6$. By lemma 2.3 (i), we obtain

$$\begin{aligned} 0>I_{c}=\lim _{n \rightarrow \infty } I\left( u_{n}\right) \ge -\limsup _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) \ge 0, \end{aligned}$$

which is a contradiction. So $\delta >0$, there exists a sequence $\left\{ y_{n}\right\} \subset \mathbb {R}^{3}$ such that $\tilde{u}_{n}(x):= u_{n}\left( x+y_{n}\right)$, then $\Vert \tilde{u}_{n}\Vert _{2}^{2}>0$. Moreover, by the translation invariance, we see that $I\left( \tilde{u}_{n}\right) \rightarrow I_{c}$ and $\left\{ \tilde{u}_{n}\right\}$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$. Then there exists $\tilde{u} \in H^{1}\left( \mathbb {R}^{3}\right)$ such that $n \rightarrow \infty$,

$$\begin{aligned} \tilde{u}_{n} \rightharpoonup \tilde{u}, \text{ in } H^{1}\left( \mathbb {R}^{3}\right) ,\quad \tilde{u}_{n}(x) \rightarrow \tilde{u}(x), \text{ a.e. } \text{ in } \mathbb {R}^{3}, \end{aligned}$$

(3.1)

which implies that $\tilde{u} \ne 0$. Let $\alpha :=\Vert \tilde{u}\Vert _{2}^{2}$, then $\alpha \in (0, c]$. We will show that $\alpha =c$. If not, then $\alpha <c$. By (3.1) and the Brézis-Lieb Lemma (see Lemma 3.5 in [34]), we have

$$\begin{aligned} \Vert \tilde{u}_{n}\Vert _{2}^{2}=\Vert \tilde{u}_{n}-\tilde{u}\Vert _{2}^{2}+\Vert \tilde{u}\Vert _{2}^{2}+o_{n}(1), \end{aligned}$$

then $\lim _{n \rightarrow \infty }\Vert \tilde{u}_{n}-\tilde{u}\Vert _{2}^{2}=c-\alpha >0$. By (3.1), the Brézis-Lieb Lemma and the splitting result Lemma 2.6 in [35], we see that

$$\begin{aligned} I_{c}=\lim _{n \rightarrow \infty } I\left( \tilde{u}_{n}\right) \ge I(\tilde{u})+\lim _{n \rightarrow \infty } I\left( \tilde{u}_{n}-\tilde{u}\right) \ge I_{\alpha }+I_{c-\alpha }, \end{aligned}$$

which contradicts to Lemma 3.4. So $\Vert \tilde{u}\Vert _{2}^{2}=c$ and subsequently,

$$\begin{aligned} I_{c} \le I(\tilde{u}) \le \lim _{n \rightarrow \infty } I\left( \tilde{u}_{n}\right) = I_{c}. \end{aligned}$$

Therefore, $\tilde{u} \in S_{c}$ serves as a minimizer of $I_{c}$.

(ii) The proof will be divided into three steps.

Step 1: If $(f_{4})$ holds, then $c^{*}>0$.

It follows from (2.4) and the Gagliardo Nirenberg inequality that

$$\begin{aligned} \begin{aligned} I(u) =&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4} -\int _{\mathbb {R}^{3}} F(u) d x \\ \ge&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4} - \epsilon \Vert u\Vert _{\frac{10}{3}}^{\frac{10}{3}}- C_{\epsilon } \Vert u\Vert _{\frac{14}{3}}^{\frac{14}{3}}\\ \ge&\frac{a}{2}\Vert \nabla u\Vert _{2}^{2}+ \frac{b}{4}\Vert \nabla u\Vert _{2}^{4}-\epsilon C c^{\frac{2}{3}} \Vert \nabla u\Vert _{2}^{2}-C_{\epsilon }Cc^{\frac{1}{3}} \Vert \nabla u\Vert _{2}^{4}\\ =&(\frac{a}{2}-\epsilon C c^{\frac{2}{3}} )\Vert \nabla u\Vert _{2}^{2}+(\frac{b}{4}-C_{\epsilon }Cc^{\frac{1}{3}} )\Vert \nabla u\Vert _{2}^{4}\ge \frac{a}{4}\Vert \nabla u\Vert _{2}^{2}, \end{aligned} \end{aligned}$$

where c is suitable small. We conclude that $I_{c} \ge 0$ for $c>0$ small. By Lemma 3.3, one has that $I_{c}=0$ for c suitable small. So $c^{*}>0$.

Step 2: $I_{c}$ has no minimizer for each $0<c<c^{*}$.

By contradiction, for some $c \in \left( 0, c^{*}\right)$, there exists a $u_{c} \in S_{c}$ which is the minimizer of $I_{c}$. Let $u_{c^{*}}:=u_{c}\left( \left( \frac{c^{*}}{c}\right) ^{-\frac{1}{3}} x\right)$, then $u_{c^{*}} \in S_{c^{*}}$. By the fact of $I\left( u_{c}\right) =I_{c}=0$, we get

$$\begin{aligned} \begin{aligned} I_{c^{*}} \le I\left( u_{c^{*}}\right)&=\left( \frac{c^{*}}{c}\right) ^{\frac{1}{3}} \frac{a}{2} \Vert \nabla u_{c}\Vert _{2}^{2}+\left( \frac{c^{*}}{c}\right) ^{\frac{2}{3}} \frac{b}{4}\Vert \nabla u_{c}\Vert _{2}^{4}-\left( \frac{c^{*}}{c}\right) \int _{\mathbb {R}^{3}} F\left( u_{c}\right) \\&=\left( \frac{c^{*}}{c}\right) \left[ \left( \left( \frac{c^{*}}{c}\right) ^{-\frac{2}{3}}-1\right) \frac{a}{2} \Vert \nabla u_{c}\Vert _{2}^{2}+\left( \left( \frac{c^{*}}{c}\right) ^{-\frac{1}{3}}-1\right) \frac{b}{4}\Vert \nabla u_{c}\Vert _{2}^{4}\right] <0, \end{aligned} \end{aligned}$$

which contradicts to $I_{c^{*}}=0$. So $I_{c}$ has no minimizer for all $c \in \left( 0, c^{*}\right)$.

Step 3: $I_{c^{*}}$ has a minimizer which is an energy ground state solution of problem (P) with the associated Lagrange multiplier being negative.

Let $c_{k}:=c^{*}+\frac{1}{k}$ for any $k \in \mathbb {N}^{+}$. Since $I_{c_{k}}<0$ by Lemma 3.3, one may choose $v_{k} \in S_{c_{k}}$ such that

$$\begin{aligned} I_{c_{k}} \le I\left( v_{k}\right) \le \frac{1}{2} I_{c_{k}}<0 \text{ for } \text{ each } k \in \mathbb {N}^{+}. \end{aligned}$$

(3.2)

Hence, $\left\| \nabla v_{k}\right\| _{2}^{2}>4\rho (c^{*})>0$, the sequence $\left\{ v_{k}\right\}$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$ by Lemma 2.3 (ii), (iii), and there exists a subsequence such that

$$\begin{aligned} \lim _{k \rightarrow \infty } \Vert \nabla v_{k}\Vert _{2} \quad \text {and}\quad \lim _{k \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( v_{k}\right) d x \quad \text {exist}. \end{aligned}$$

Next, we prove that $\left\{ v_{k}\right\}$ is non-vanishing, that is

$$\begin{aligned} \sigma :=\lim _{k \rightarrow \infty }\sup _{y \in \mathbb {R}^{3}} \int _{B(y, 1)}\left| v_{k}\right| ^{2} d x>0. \end{aligned}$$

Otherwise, $\sigma =0$ and the vanishing lemma gives that $\lim _{k \rightarrow \infty }\left\| v_{k}\right\| _{\frac{14}{3}} \rightarrow 0$. By Lemma 2.3 (i) and $\lim _{k \rightarrow \infty }\left\| \nabla v_{k}\right\| _{2}^{2}>0$, we obtain

$$\begin{aligned} \lim _{k \rightarrow \infty } I\left( v_{k}\right) \ge \frac{a}{2} \lim _{k \rightarrow \infty }\left\| \nabla v_{k}\right\| _{2}^{2}+\frac{b}{4} \lim _{k \rightarrow \infty }\left\| \nabla v_{k}\right\| _{2}^{4}>0. \end{aligned}$$

This contradicts to (3.2) and hence $\sigma >0$. Since $\left\{ v_{k}\right\}$ is non-vanishing, there exists a sequence $\left\{ y_{k}\right\} \subset \mathbb {R}^{3}$ and a nontrivial function $v \in H^{1}\left( \mathbb {R}^{3}\right)$ such that up to a subsequence $v_{k}\left( \cdot +y_{k}\right) \rightharpoonup v$ in $H^{1}\left( \mathbb {R}^{3}\right)$ and $v_{k}\left( \cdot +y_{k}\right) \rightarrow v$ almost everywhere on $\mathbb {R}^{3}$.

Denote $c^{\prime }:=\Vert v\Vert _{2}^{2} \in \left( 0, c^{*}\right]$ and $w_{k}:=v_{k}\left( \cdot +y_{k}\right) -v$. Clearly, $\lim _{k \rightarrow \infty }\left\| w_{k}\right\| _{2}^{2}=c^{*}-c^{\prime }$ and using the Brézis-Lieb Lemma, we have

$$\begin{aligned} \lim _{k \rightarrow \infty }\left\| \nabla v_{k}\right\| _{2}^{2}=\Vert \nabla v\Vert _{2}^{2}+\lim _{k \rightarrow \infty }\left\| \nabla w_{k}\right\| _{2}^{2}. \end{aligned}$$

Noting that $I_{c_{k}} \rightarrow I_{c^{*}}=0$ by Lemma 3.2, we deduce from (3.2) and the splitting result Lemma 2.6 in [35] that

$$\begin{aligned} 0=\lim _{k \rightarrow \infty } I\left( v_{k}\right) =\lim _{k \rightarrow \infty } I\left( v+w_{k}\right) \ge I(v)+\lim _{k \rightarrow \infty } I\left( w_{k}\right) . \end{aligned}$$

(3.3)

Since $I_{s}=0$ for any $s \in \left( 0, c^{*}\right]$ by Step 1, therefore $I(v) \ge 0$ and $\lim _{k \rightarrow \infty } I\left( w_{k}\right) \ge 0$. In view of (3.3), we obtain

$$\begin{aligned} I(v)=0 \quad \text{ and } \quad \lim _{k \rightarrow \infty } I\left( w_{k}\right) =0. \end{aligned}$$

Now, from the fact that Step 2, the global infimum $I_{s}=0$ is not achieved for any $s \in \left( 0, c^{*}\right)$, it follows that $c^{\prime }=c^{*}$ and thus $v \in S_{c^{*}}$ is a minimizer of $I_{c^{*}}=0$. As a consequence, $v \in S_{c^{*}}$ is an energy ground state solution of problem (P) with a Lagrange multiplier $\lambda \in \mathbb {R}$. Using the Pohozaev identity (2.2), it is easy to see that

$$\begin{aligned} 0=I(v)=I(v)-\frac{1}{6} Q(v)=\frac{a}{3}\Vert \nabla v\Vert _{2}^{2}+\frac{b}{12}\Vert \nabla v\Vert _{2}^{4}+\frac{1}{2}\lambda c^{*}, \end{aligned}$$

and hence $\lambda <0$. The proof of Theorem 1.1 is complete.

4 The Proof of Theorem 1.2

In this section, we will first explore the properties of the local infimum $\bar{I}_{c}$ in subsection 4.1. Then, in subsection 4.2, we will proceed to complete the proof of Theorem 1.2. Recalling the definition, for any $c\in \left( 0, c^{*}\right]$,

$$\begin{aligned} \bar{I}_{c}=\inf _{u \in S_{c}^{\rho }} I(u) \ge \inf _{u \in S_{c}}I(u)=0, \end{aligned}$$

where $S_{c}^{\rho }:=\left\{ u \in S_{c} \mid \Vert \nabla u\Vert _{2}^{2}>\rho \left( c^{*}\right) \right\}$ and $\rho \left( c^{*}\right) >0$ is the value provided in lemma 2.3 (iii).

4.1 Properties of the Local Infimum

Lemma 4.1

Assume that f satisfies $(f_0)-(f_4)$. Then the following conclusions are true.

$\mathrm{(i)}$:

 The function $c \mapsto \bar{I}_{c}$ is continuous on $\left( 0, c^{*}\right] ;$

$\mathrm{(ii)}$:

 The function $c \mapsto \bar{I}_{c}$ is non-increasing on $\left( 0, c^{*}\right] ;$

Proof

(i) It is sufficient to prove that for a given $c \in \left( 0, c^{*}\right]$ and any sequence $\left\{ c_{n}\right\} \subset \left( 0, c^{*}\right)$ such that $c_{n} \rightarrow c$ as $n \rightarrow \infty$ one has $\lim _{n \rightarrow \infty } \bar{I}_{c_{n}}=\bar{I}_{c}$. Noting that for any $u \in S_{c}^{\rho }$, set $u_{n}:=\sqrt{\frac{c_{n}}{c} } \cdot u \rightarrow u$ in $H^{1}\left( \mathbb {R}^{3}\right)$ as $n \rightarrow \infty$. Then, using the fact that $u_{n} \in S_{c_{n}}^{\rho }$ and $\lim _{n \rightarrow \infty } I\left( u_{n}\right) =I(u)$. Thus

$$\begin{aligned} \limsup _{n \rightarrow \infty } \bar{I}_{c_{n}} \le \limsup _{n \rightarrow \infty } I(u_{n})=I(u). \end{aligned}$$

(4.1)

By the arbitrariness of $u \in S_{c}^{\rho }$, we conclude that

$$\begin{aligned} \limsup _{n \rightarrow \infty } \bar{I}_{c_{n}} \le \bar{I}_{c}. \end{aligned}$$

(4.2)

To complete the proof, it remains to show

$$\begin{aligned} \liminf _{n \rightarrow \infty } \bar{I}_{c_{n}} \ge \bar{I}_{c}. \end{aligned}$$

(4.3)

From the definition of $\bar{I}_{c_{n}}$, for each $n \in \mathbb {N}^{+}$, there exists $v_{n} \in S_{c_{n}}^{\rho }$ such that

$$\begin{aligned} I\left( v_{n}\right) \le \bar{I}_{c_{n}}+\frac{1}{n}. \end{aligned}$$

(4.4)

Setting $t_{n}:=\sqrt{\frac{c}{c_{n}}}$, we have $\tilde{v}_{n}:=t_{n}^{-\frac{1}{2}} v_{n}(\frac{\cdot }{t_{n}}) \in S_{c}^{\rho }$ and thus

$$\begin{aligned} \bar{I}_{c}&\le I\left( \tilde{v}_{n}\right) \le I\left( v_{n}\right) +\left| I\left( \tilde{v}_{n}\right) -I\left( v_{n}\right) \right| \le \bar{I}_{c_{n}}+\frac{1}{n}+\left| I\left( \tilde{v}_{n}\right) -I\left( v_{n}\right) \right| \\&\le \bar{I}_{c_{n}}+\frac{1}{n}+t_{n}^{3} \int _{\mathbb {R}^{3}}\left| F\left( t_{n}^{-\frac{1}{2}} v_{n}\right) -F\left( v_{n}\right) \right| d x+\left| t_{n}^{3}-1\right| \int _{\mathbb {R}^{3}}\left| F\left( v_{n}\right) \right| d x. \end{aligned}$$

Since $t_{n} \rightarrow 1$ and f satisfies $(f_1)$, $(f_2)$, to prove (4.3) is equivalent to show $\left\{ v_{n}\right\}$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$. To substantiate the boundedness, we can reference (4.2) and (4.4), which yields $\limsup _{n \rightarrow \infty } I\left( v_{n}\right) \le \bar{I}_{c}$. Additionally, it is worth noting that as $v_{n} \in S_{c_{n}}^{\rho }$ and $c_{n} \rightarrow c$, Lemma 2.3 (ii) guarantees that $\left\{ v_{n}\right\}$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$.

(ii) Let $0<c^{\prime }<c\le c^{*}$. It is equivalent to show that for an arbitrary $\varepsilon >0$ one has

$$\begin{aligned} \bar{I}_{c} \le \bar{I}_{c^{\prime }}+\varepsilon . \end{aligned}$$

(4.5)

From the definition of $\bar{I}_{c^{\prime }}$ and functional I is coercive, there exist $u \in S_{c^{\prime }}^{\rho }$, constant $M>0$, such that

$$\begin{aligned} I(u) \le \bar{I}_{c^{\prime }}+\frac{\varepsilon }{2}, \quad \rho \left( c^{*}\right) <\Vert \nabla u\Vert _{2}^{2}\le M. \end{aligned}$$

(4.6)

We adopt some ideas from [25]. Let $\tau \in C_{c}^{\infty }\left( \mathbb {R}^{3}\right)$ be a radial cut-off function such that

$$\begin{aligned} \tau (x)=\left\{ \begin{array}{lll} 1 &{} \text{ if } &{} |x| \le 1, \\ 0 &{} \text{ if } &{} |x| \ge 2. \end{array}\right. \end{aligned}$$

For any $\delta >0$, we set $u_{\delta }(x):=u(x) \tau (\delta x)$. Since $u_{\delta } \rightarrow u$ in $H^{1}\left( \mathbb {R}^{3}\right)$ as $\delta \rightarrow 0^{+}$, one can fix a small enough constant $\delta >0$ such that $\rho \left( c^{*}\right) <\left\| \nabla u_{\delta }\right\| _{2}^{2}\le M$ and

$$\begin{aligned} I\left( u_{\delta }\right) \le I(u)+\frac{\varepsilon }{4}. \end{aligned}$$

(4.7)

Taking $v \in C_{c}^{\infty }\left( \mathbb {R}^{3}\right) \backslash \{0\}$ such that ${\text {supp}}(v) \subset B(0,1+4 / \delta ) \backslash B(0,4 / \delta )$ and set

$$\begin{aligned} \tilde{v}:=\frac{\sqrt{c-\left\| u_{\delta }\right\| _{2}^{2}}}{\Vert v\Vert _{2}}v. \end{aligned}$$

For any $s \le 0$, we define $w_{s}:=u_{\delta }+s \star \tilde{v}$. Since $u_{\delta }$ and $s \star \tilde{v}$ have disjoint supports, it is clear that $w_{s} \in S_{c}^{\rho }$. Noting that $\Vert \nabla (s \star \tilde{v})\Vert _{2} \rightarrow 0$ as $s \rightarrow -\infty$, it follows from Lemma 2.3 (i) that

$$\begin{aligned} I\left( s_{0} \star \tilde{v}\right) \le \frac{\varepsilon }{8} \quad \text{ for } \text{ some } s_{0}<0 \text{ such } \text{ that } \Vert \nabla (s_{0} \star \tilde{v})\Vert _{2}^{2}\le \frac{\varepsilon }{4bM}. \end{aligned}$$

(4.8)

Now, by the definition of $\bar{I}_{c}$, (4.6)–(4.8), we obtain

$$\begin{aligned} \bar{I}_{c} \le I\left( w_{s_{0}}\right) = I\left( u_{\delta }\right) +I\left( s_{0} \star \tilde{v}\right) +\frac{b}{2}\Vert \nabla u_{\delta }\Vert _{2}^{2}\Vert \nabla (s_{0} \star \tilde{v})\Vert _{2}^{2} \le I(u)+\frac{\varepsilon }{2} \le \bar{I}_{c^{\prime }}+\varepsilon , \end{aligned}$$

that is (4.5). The proof is complete. $\square$

Lemma 4.2

Assume that f satisfies $(f_0)-(f_4)$. Suppose that for some $c^{\prime } \in \left( 0, c^{*}\right]$, there exists a couple $(u, \lambda ) \in S_{c^{\prime }}^{\rho } \times \mathbb {R}$ such that

$$\begin{aligned} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u), \end{aligned}$$

and $I(u)=\bar{I}_{c^{\prime }}$. Then $\bar{I}_{c}<\bar{I}_{c^{\prime }}$ for any $c<c^{\prime }$ close enough to $c^{\prime }$ if $\lambda >0$ and for each $c \in \left( c^{\prime }, c^{*}\right]$ near enough to $c^{\prime }$ if $\lambda <0$.

Proof

We only consider the case that $\lambda < 0$ and the situation of $\lambda >0$ follows analogously. Let $u \in S_{c^{\prime }}^{\rho }$ and $\lambda \in \mathbb {R}$ be as above. For any $t>0$, we set $u_{t}:=t u \in S_{c^{\prime } t^{2}}$ and

$$\begin{aligned} \alpha (t):=I\left( u_{t}\right) =\frac{a}{2} t^{2} \Vert \nabla u\Vert _{2}^{2}+\frac{b}{4} t^{4} \Vert \nabla u\Vert _{2}^{4}-\int _{\mathbb {R}^{3}} F(t u) d x. \end{aligned}$$

If $\lambda <0$, noting that $u_{t} \rightarrow u$ strongly in $H^{1}\left( \mathbb {R}^{3}\right)$ as $t \rightarrow 1$ and the fact that

$$\begin{aligned} \left\langle I^{\prime }(u), u\right\rangle = \lambda \Vert u\Vert _{2}^{2}=\lambda c^{\prime }<0. \end{aligned}$$

There exists a $\delta >0$ small enough such that for any $t \in [1, 1+\delta )$,

$$\begin{aligned} u_{t} \in S_{c^{\prime } t^{2}}^{\rho } \quad \text{ and } \quad \frac{d}{d t} \alpha (t)=t^{-1} I^{\prime }\left( u_{t}\right) u_{t}<0. \end{aligned}$$

Consequently, from the mean value theorem, it follows that

$$\begin{aligned} \bar{I}_{c^{\prime } t^{2}}\le \alpha (t)=\alpha (1)+(t-1) \cdot \frac{d}{d t} \alpha (\theta )<\alpha (1), \end{aligned}$$

where $1<\theta<t<1+\delta$. For any $c^{\prime }<c$ close enough to $c^{\prime }$, we have $t:=\sqrt{\frac{c}{c^{\prime }}} \in [1,1+\delta )$ and thus $\alpha (1)=I(u)=\bar{I}_{c^{\prime }}>\bar{I}_{c}$. The case of $\lambda >0$ can be proved similarly.

$\square$

As a direct consequence, from lemmas 4.1, 4.2, we directly obtain the following lemma which plays a crucial role in investigating the convergence of the Palais-Smale sequences at the suspected ground state energy level $\bar{I}_{c}$.

Lemma 4.3

Assume f satisfies $(f_0)-(f_4)$. If for some $c^{\prime } \in \left( 0, c^{*}\right]$ there exists a couple $(u, \lambda ) \in S_{c^{\prime }}^{\rho } \times \mathbb {R}$ such that

$$\begin{aligned} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u), \end{aligned}$$

and $I(u)=\bar{I}_{c^{\prime }}$, then $\lambda \le 0$. If in addition $\lambda <0$, then $\bar{I}_{c}<\bar{I}_{c^{\prime }}$ for any $c \in \left( c^{\prime }, c^{*}\right]$.

4.2 Existence of Local Minimizers

In this subsection, we will establish the existence of local minimizers for a suitable range of the mass and then finish the proof of Theorem 1.2. We demonstrate at first the following geometrical result about the local infimum $\bar{I}_{c}$.

Lemma 4.4

Assume that f satisfies $(f_0)-(f_4)$. Then there exists $c^{* *} \in \left( 0, c^{*}\right)$ such that for any $c \in \left( c^{* *}, c^{*}\right]$

$$\begin{aligned} \bar{I}_{c}:=\inf _{u \in S_{c}^{\rho }} I(u)<\frac{a}{4}\rho \left( c^{*}\right) \le \inf _{u \in \mathcal {M}_{c}} I(u), \end{aligned}$$

where $\mathcal {M}_{c}:=\left\{ u \in S_{c} \mid \rho \left( c^{*}\right) <\Vert \nabla u\Vert _{2}^{2} \le 4 \rho \left( c^{*}\right) \right\}$. In particular, $\bar{I}_{c^{*}}=0$ is achieved.

Proof

By Theorem 1.1, we can fix a minimizer $v \in S_{c^{*}}$ of $I_{c^{*}}=0$. From Lemma 2.3 (iii), it follows that

$$\begin{aligned} \Vert \nabla v\Vert _{2}^{2}>4 \rho \left( c^{*}\right) ,\quad \inf _{u \in \mathcal {M}_{c}} I(u) \ge \frac{a}{4}\rho \left( c^{*}\right) \quad \text{ and } \quad I( v)=0. \end{aligned}$$

Then $v \in S_{c^{*}}^{\rho }$ and $\bar{I}_{c^{*}}=0$ is achieved. Moreover, by continuity there exists $z \in (0,1)$ such that, for any $t \in (z, 1]$,

$$\begin{aligned} \Vert \nabla (t v)\Vert _{2}^{2}>4\rho \left( c^{*}\right) \quad \text{ and } \quad I(t v)<\frac{a}{4}\rho \left( c^{*}\right) . \end{aligned}$$

Clearly, the proof is complete with the choice of $c^{* *}:=z^{2} c^{*}$. $\square$

Remark 4.1

Lemma 2.3 (iii) gives that $\Vert \nabla u\Vert _{2}^{2}>4 \rho \left( c^{*}\right)$ for any $u \in S_{c}^{\rho }$ with $I(u)< \frac{a}{4}\rho \left( c^{*}\right)$, and hence an arbitrary minimizing sequence $\left\{ u_{n}\right\} \subset S_{c}^{\rho }$ of $\bar{I}_{c}$ satisfies

$$\begin{aligned} \left\| \nabla u_{n}\right\| _{2}^{2}>4 \rho \left( c^{*}\right) \text{ for } \text{ any } n \text{ large } \text{ enough. } \end{aligned}$$

In Lemma 4.5, it becomes evident that this additional information is crucial in overcoming certain challenges posed by the local constraint $\Vert \nabla u\Vert _{2}^{2}>\rho \left( c^{*}\right)$ on $S_{c}$. Furthermore, the upper estimate $\bar{I}_{c}<\frac{a}{4}\rho \left( c^{*}\right)$ allows us to demonstrate that the Lagrange multiplier is strictly negative in Lemma 4.5. This is of significant importance in our compactness argument when seeking minimizers of the local infimum $\bar{I}_{c}$.

To tackle the compactness issue when seeking minimizers for $\bar{I}_{c}$, we present a compactness lemma.

Lemma 4.5

Assume that f satisfies $(f_0)-(f_4)$. Let $c \in \left( c^{* *}, c^{*}\right]$ and $\left\{ u_{n}\right\} \subset S_{c}^{\rho }$ be an arbitrary minimizing sequence of $\bar{I}_{c}$. Then there exist $\{y_{n}\}\subset \mathbb {R}^{3}$, $u \in S_{c}^{\rho }$ such that $u_{n}\left( \cdot +y_{n}\right) \rightarrow u$ in $H^{1}\left( \mathbb {R}^{3}\right)$. In particular, u is a minimizer of the local infimum $\bar{I}_{c}$ with

$$\begin{aligned} I(u)=\left\{ \begin{array}{ll} \bar{I}_{c}>0 &{} \text{ for } c \in \left( c^{* *}, c^{*}\right) , \\ \bar{I}_{c}=\inf _{u \in S_{c}} I(u)=0 &{} \text{ for } c=c^{*}, \end{array}\right. \end{aligned}$$

(4.9)

and the associated Lagrange multiplier being negative.

Proof

Since $\left\{ u_{n}\right\} \subset S_{c}^{\rho }$ is bounded in $H^{1}\left( \mathbb {R}^{3}\right)$ by Lemma 2.3 (ii). Thus one may assume that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert \nabla u_{n}\Vert _{2}^{2}\quad \text {and} ~\lim _{n \rightarrow \infty } \int _{\mathbb {R}^{3}} F\left( u_{n}\right) d x~ \text {exist}. \end{aligned}$$

We claim that $\left\{ u_{n}\right\}$ is non-vanishing, that is $\sigma :=\lim _{n \rightarrow \infty }\sup _{y \in \mathbb {R}^{3}} \int _{B(y, 1)}\left| u_{n}\right| ^{2} d x>0$. If not, i.e. $\sigma =0$, the vanishing lemma imply that $\lim _{n \rightarrow \infty }\left\| u_{n}\right\| _{\frac{14}{3}} = 0$. In view of Lemma 2.3 (i), we have

$$\begin{aligned} \bar{I}_{c}=\lim _{n \rightarrow \infty } I\left( u_{n}\right) \ge \frac{a}{2} \lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}^{2} \ge \frac{a}{2} \rho \left( c^{*}\right) . \end{aligned}$$

This contradicts to Lemma 4.4 thereby establishing the claim. Then, there exists a sequence $\left\{ y_{n}\right\} \subset \mathbb {R}^{3}$ and a nontrivial $u \in H^{1}\left( \mathbb {R}^{3}\right)$ such that up to a subsequence $\bar{u}_{n}:=u_{n}\left( \cdot +y_{n}\right) \rightharpoonup u$ in $H^{1}\left( \mathbb {R}^{3}\right)$ and $\bar{u}_{n} \rightarrow u$ almost everywhere on $\mathbb {R}^{3}$.

Denote $c^{\prime }:=\Vert u\Vert _{2}^{2} \in (0, c]$ and $w_{n}:=\bar{u}_{n}-u$. It is clear that $\lim _{n \rightarrow \infty }\left\| w_{n}\right\| _{2}^{2}=c-c^{\prime }$,

$$\begin{aligned} \lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}^{2}=\Vert \nabla u\Vert _{2}^{2}+\lim _{n \rightarrow \infty }\left\| \nabla w_{n}\right\| _{2}^{2}, \end{aligned}$$

(4.10)

and by the splitting result Lemma 2.6 in [35] and Lemma 2.10 in [36], we obtain

$$\begin{aligned} \bar{I}_{c}=\lim _{n \rightarrow \infty } I\left( u_{n}\right) =\lim _{n \rightarrow \infty } I\left( u+w_{n}\right) = I(u)+\lim _{n \rightarrow \infty }\left( I\left( w_{n}\right) +\frac{b\Vert \nabla w_{n}\Vert _{2}^{2}\Vert \nabla u\Vert _{2}^{2}}{2}\right) .\nonumber \\ \end{aligned}$$

(4.11)

The next step involves determining the weak limit $u \in S_{c^{\prime }}$ on the local constraint $S_{c^{\prime }}^{\rho }$. Precisely, we aim to demonstrate that

$$\begin{aligned} \Vert \nabla u\Vert _{2}^{2}>\rho \left( c^{*}\right) . \end{aligned}$$

(4.12)

If (4.12) is false, then $\Vert \nabla u\Vert _{2}^{2} \le \rho \left( c^{*}\right)$. Since $\lim _{n \rightarrow \infty }\left\| \nabla u_{n}\right\| _{2}^{2} \ge 4\rho \left( c^{*}\right)$ by Lemma 4.4, it follows from (4.10) that

$$\begin{aligned} \left\| \nabla w_{n}\right\| _{2}^{2} \ge 3\rho \left( c^{*}\right) >\rho \left( c^{*}\right) \text{ for } \text{ any } n \text{ large } \text{ enough. } \end{aligned}$$

(4.13)

To arrive at a contradiction, we differentiate between two cases: compactness and non-compactness.

•  Compactness: that is $c^{\prime }=c$. Then $\lim _{n \rightarrow \infty }\left\| w_{n}\right\| _{\frac{14}{3}} = 0$ by the Gagliardo Nirenberg inequality. In view of Lemma 2.3 (i) and (4.13), it is evident that

  $$\begin{aligned} \lim _{n \rightarrow \infty } I\left( w_{n}\right) \ge \frac{a}{2} \lim _{n \rightarrow \infty }\left\| \nabla w_{n}\right\| _{2}^{2} \ge \frac{a}{2}\rho \left( c^{*}\right) . \end{aligned}$$

  Using (4.11) and $I(u) \ge I_{c^{\prime }}=0$ by Theorem 1.1, we have

  $$\begin{aligned} \bar{I}_{c}\ge I(u)+\lim _{n \rightarrow \infty } I\left( w_{n}\right) \ge \frac{a}{2}\rho \left( c^{*}\right) , \end{aligned}$$

  which contradicts Lemma 4.4.

•  Non-compactness: that is $c^{\prime }<c$. In this case, $t_{n}:=\left\| w_{n}\right\| _{2}^{2} \rightarrow c-c^{\prime } \in (0, c)$ as $n\rightarrow \infty$ and $w_{n} \in S_{t_{n}}^{\rho }$ for any n large enough by (4.13). Using the fact that $\bar{I}_{c}$ is non-increasing by Lemma 4.1, we have

  $$\begin{aligned} \lim _{n \rightarrow \infty } I\left( w_{n}\right) \ge \limsup \bar{I}_{t_{n}} \ge \bar{I}_{c}. \end{aligned}$$

  Since $I_{c^{\prime }}=0$ is not achieved by Theorem 1.1, it follows that $I(u)>I_{c^{\prime }}=0$. In view of (4.11), we obtain

  $$\begin{aligned} \bar{I}_{c}\ge I(u)+\lim _{n \rightarrow \infty } I\left( w_{n}\right) >\bar{I}_{c}, \end{aligned}$$

  which is a contradiction.

With the desired location estimate (4.12) at hand, we directly obtain $I(u) \ge \bar{I}_{c^{\prime }}$.

Since $I_{s}=0$ for any $s \in \left( 0, c^{*}\right]$ by Theorem 1.1, thus $\lim _{n \rightarrow \infty } I\left( w_{n}\right) \ge 0$. Using (4.11) and Lemma 4.1, we know

$$\begin{aligned} \bar{I}_{c}&=\lim _{n \rightarrow \infty } I\left( u+w_{n}\right) = I(u)+\lim _{n \rightarrow \infty }\left( I\left( w_{n}\right) +\frac{b\Vert \nabla w_{n}\Vert _{2}^{2}\Vert \nabla u\Vert _{2}^{2}}{2}\right) \\&\ge \bar{I}_{c^{\prime }}+\lim _{n \rightarrow \infty }\left( I\left( w_{n}\right) +\frac{b\Vert \nabla w_{n}\Vert _{2}^{2}\Vert \nabla u\Vert _{2}^{2}}{2}\right) \ge \bar{I}_{c'}\ge \bar{I}_{c}. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \lim _{n \rightarrow \infty } I\left( w_{n}\right) =0,\quad \lim _{n \rightarrow \infty }\Vert \nabla w_{n}\Vert _{2}^{2}=0 \quad \text {and}\quad \bar{I}_{c}=I(u)=\bar{I}_{c^{\prime }}. \end{aligned}$$

(4.14)

By (4.10), we know that $\lim _{n \rightarrow \infty }\Vert \nabla u_{n}\Vert _{2}^{2}=\Vert \nabla u\Vert _{2}^{2}$. In particular, the weak limit $u \in S_{c^{\prime }}^{\rho }$ is a minimizer of $\bar{I}_{c^{\prime }}$ and hence there exists a Lagrange multiplier $\lambda \in \mathbb {R}$ such that

$$\begin{aligned} -\left( a+b \int _{\mathbb {R}^{3}}|\nabla u|^{2}dx\right) \Delta u-\lambda u=f(u) \end{aligned}$$

(4.15)

and also $Q(u)=0$, where Q is defined in (2.2). Then, by Lemma 4.4 and (4.14),

$$\begin{aligned} \frac{a}{4} \rho \left( c^{*}\right) >\bar{I}_{c}=I(u)=I(u)-\frac{1}{6}Q(u)=\frac{a}{3}\Vert \nabla u\Vert _{2}^{2}+\frac{b}{12}\Vert \nabla u\Vert _{2}^{4}+\frac{1}{2} \lambda \Vert u\Vert _{2}^{2}. \end{aligned}$$

Taking (4.12) into account we conclude that $\lambda <0$. Now, in view of (4.14), (4.15) and Lemma 4.3, it is clear that $c^{\prime }=c$ and thus $u \in S_{c}^{\rho }$ is a minimizer of $\bar{I}_{c}$. In particular, (4.9) follows from the facts that $I_{c}=0$ is not achieved when $c \in \left( 0, c^{*}\right)$ and that $\bar{I}_{c^{*}}=0$ is achieved by lemma 4.4. In view of $\lim _{n \rightarrow \infty }\Vert \nabla w_{n}\Vert _{2}^{2}=0$, and $c=c^{\prime }$, hence $\bar{u}_{n} \rightarrow u \in S_{c}^{\rho }$ in $H^{1}\left( \mathbb {R}^{3}\right)$. $\square$

Proof of Theorem 1.2

Considering $c^{* *} \in \left( 0, c^{*}\right)$ as defined in lemma 4.4. All the conclusions of Theorem 1.2 can be deduced from Lemma 4.5. We only need to prove solution u is an energy ground state solution of problem (P) when (1.6) holds. By the fact that u satisfies the Pohozaev identity $P(u)=0$, we can conclude that u belongs to $S_{c}^{\rho }$ since Lemma 2.3 (iii). Hence, the local minimizer $u\in S_{c}^{\rho }$ is actually an energy ground state solution of (P). $\square$

5 The Proof of Theorems 1.3 and 1.4

This section is dedicated to the proof of Theorem 1.3 by using mountain pass arguments. To simplify, we define $\bar{\rho }(c):=\rho (c^{*})$ for $c \in (0, c^{*}]$ and $\bar{\rho }(c):=\rho (c)$ if $c> c^{*}$, where $\rho (c)>0$ being the value provided in Lemma 2.3 (iii). For sufficiently large value of $m>1$ such that

$$\begin{aligned} \frac{a}{4}\bar{\rho }(c)-\frac{b\bar{\rho }(c)^{2}}{4m^{2}}>0, \end{aligned}$$

we introduce the set of continuous paths

$$\begin{aligned} \Gamma _{c}:=\left\{ \gamma \in C\left( [0,1], S_{c,r}\right) \Bigg | \begin{array}{l} \Vert \nabla \gamma (0)\Vert _{2}^{2}<\bar{\rho }(c), \\ \Vert \nabla \gamma (1)\Vert _{2}^{2}>4 \bar{\rho }(c), \\ \max \{I(\gamma (0)), I(\gamma (1))\}<(\frac{a}{4}+ \frac{a}{2m})\bar{\rho }(c) \end{array}\right\} . \end{aligned}$$

For our subsequent discussions, we establish the non-emptiness of $\Gamma _{c}$ in the following lemma.

Lemma 5.1

Assume that f satisfies $(f_0)-(f_4)$. Then $\Gamma _{c} \ne \emptyset$ for any $c>c^{* *}$.

Proof

Initially, we observe that if we find $u \in S_{c,r}$ that fulfills

$$\begin{aligned} \Vert \nabla u\Vert _{2}^{2}>4 \bar{\rho }(c) \quad \text{ and } \quad I(u)<\left( \frac{a}{4}+ \frac{a}{2m}\right) \bar{\rho }(c), \end{aligned}$$

(5.1)

then $\Gamma _{c} \ne \emptyset$. Indeed, by Lemma 2.3 (i), for such $u \in S_{c,r}$, given that $\Vert \nabla (s \star u) \Vert _{2}$ converges to 0 as $s \rightarrow -\infty$, we can choose $s<0$ such that

$$\begin{aligned} \Vert \nabla (s\star u)\Vert _{2}^{2}<\frac{1}{m} \bar{\rho }(c), \quad \left| \int _{\mathbb {R}^{3}} F(s\star u) d x\right| <\frac{a}{4}\bar{\rho }(c)-\frac{b\bar{\rho }(c)^{2}}{4m^{2}}. \end{aligned}$$

Consequently, we have

$$\begin{aligned} I(s \star u)<(\frac{a}{4}+ \frac{a}{2m})\bar{\rho }(c). \end{aligned}$$

Setting $\gamma (t):=(s(1-t)) \star u$ for any $t \in [0,1]$ and combining (5.1), we conclude that $\gamma \in \Gamma _{c}$.

To complete this lemma, we need to find a function $u \in S_{c,r}$ that satisfies (5.1). For $c \ge c^{*}$, by Theorem 1.1, the constrained functional $I_{\mid S_{c}}$ admits a global minimizer $v \in S_{c}$ with $I(v) \le 0$. Referring to Theorem 2 in [37], this minimizer is radially symmetric up to a translation in $\mathbb {R}^{3}$. Therefore, we can assume that $v \in S_{c,r}$. Since $I(v) \le 0$, we can deduce from Lemma 2.3 (iii) that $\Vert \nabla v\Vert _{2}^{2}>4 \bar{\rho }(c)$, which implies that $u:=v \in S_{c,r}$ satisfies (5.1). In the case that $c \in \left( c^{* *}, c^{*}\right)$, we can assert that $u:=\sqrt{\frac{c}{c^{*}}} \cdot v \in S_{c,r}$ satisfies (5.1) since the global minimizer $v \in S_{c^{*}}$ is radial and the definition of $m^{**}$. $\square$

Lemma 5.2

Assume that f satisfies $(f_0)-(f_4)$. For any $c>c^{* *}$, we define the mountain pass value

$$\begin{aligned} m_{p}:=\inf _{\gamma \in \Gamma _{c}} \max _{t \in [0,1]} I(\gamma (t)). \end{aligned}$$

Then $m_{p} \ge a\bar{\rho }(c)>0$.

Proof

Since $\Gamma _{c}$ is nonempty by Lemma 5.1, we can establish the existence of the mountain pass value $m_{p}$. For any $\gamma \in \Gamma _{c}$, it holds that

$$\begin{aligned} \Vert \nabla \gamma (0)\Vert _{2}^{2}<\bar{\rho }(c) \quad \text{ and } \quad \Vert \nabla \gamma (1)\Vert _{2}^{2}>4 \bar{\rho }(c), \end{aligned}$$

according to the intermediate value theorem, there exists a $s \in (0,1)$ such that

$$\begin{aligned} \Vert \nabla \gamma (s)\Vert _{2}^{2}=4\bar{\rho }(c). \end{aligned}$$

Considering the definition of $\bar{\rho }(c)$ and Lemma 2.3 (iii), we deduce that

$$\begin{aligned} \max _{t \in [0,1]} I(\gamma (t)) \ge I(\gamma (s)) \ge a\bar{\rho }(c), \end{aligned}$$

thus $m_{p} \ge a \bar{\rho }(c)>0$. $\square$

By employing the deformation result from Lemma 2.4 and the compactness result in Lemma 2.5, we can now obtain a normalized radial solution at the mountain pass level.

Lemma 5.3

Assume that f satisfies $(f_0)-(f_5)$. Then for each $c>c^{* *}$, the constrained functional $I_{\mid S_{c,r}}$ admits a normalized radial solution at the mountain pass level $m_{p}$.

Proof

By contradiction that $K^{m_{p}}=\emptyset$. Employing the deformation result from Lemma 2.4 with $\mathcal {O}=\emptyset$ and $\bar{\varepsilon }=\frac{3m-2}{4m}a\bar{\rho }(c)>0$ for $m>1$, there exists $\varepsilon \in (0, \bar{\varepsilon })$ and $\eta \in C\left( [0,1] \times S_{c,r}, S_{c,r}\right)$ such that

$$\begin{aligned} \eta \left( 1, I^{m_{p}+\varepsilon }\right) \subset I^{m_{p}-\varepsilon }. \end{aligned}$$

(5.2)

By the definition of $m_{p}$, we can choose $\gamma \in \Gamma _{c}$ such that

$$\begin{aligned} \max _{t \in [0,1]} I(\gamma (t)) \le m_{p}+\varepsilon . \end{aligned}$$

(5.3)

Now, considering the new path $\bar{\gamma }(t):=\eta (1, \gamma (t))$ for $t \in [0,1]$. Since

$$\begin{aligned} \max \{I(\gamma (0)), I(\gamma (1))\}<\left( \frac{a}{4}+ \frac{a}{2m}\right) \bar{\rho }(c)\le m_{p}-\bar{\varepsilon }, \end{aligned}$$

it follows from Lemma 2.4 (ii) that $\bar{\gamma } \in \Gamma _{c}$. By the definition of $m_{p}$, (5.2) and (5.3), we obtain a contradiction: $m_{p} \le \max _{t \in [0,1]} I(\bar{\gamma }(t)) \le m_{p}-\varepsilon$. $\square$

Proof of Theorem 1.3

This theorem directly follows from lemmas 5.2 and 5.3. $\square$

Proof of Theorem 1.4

Based on the Symmetric Mountain Pass Theorem developed by Jeanjean and Lu [25]. Combining with Lemma 2.4 and 2.5, we can obtain our results by the same arguments as the proof of Theorem 1.4 in [25] with using the genus theory. $\square$