Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Super-Critical Case

Abstract

In this paper, we study the existence of normalized solution to the following nonlinear mass super-critical Kirchhoff equation

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u+V(x)u+\lambda u=g(u) \ \ {in} \ {{\mathbb {R}}^{N}}\\&0\le u\in H^{1}_{r}({\mathbb {R}}^{N}) \end{aligned}\right. \end{aligned}$$

where $a ,b>0$ are constants, $\lambda \in R$, and V(x) satisfies appropriate assumptions; g has a mass super-critical growth when $N=3$, and $g(u)=|u|^{p-2}u$ with $p\in (2+\frac{8}{N},2^{*}), 2^{*}=\frac{2N}{N-2}$ when $N\ge 3$. Here, we prove the existence of ground state normalized solution via variational methods.

1 Introduction and Main Results

The aim of this paper is to study the existence of normalized solutions to the following nonlinear Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u +V(x)u+\lambda u=g(u) \ \ {in} \ {{\mathbb {R}}^{N}} \end{aligned}$$

(1.1)

satisfying a normalization constraint

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}u^{2}\textrm{d}x=c. \end{aligned}$$

(1.2)

where a, b, c are positive real numbers, and $\lambda$ is unknown and will appear as a Lagrange multiplier; V(x) satisfies appropriate assumptions; g has a mass super-critical grow when $N=3$, and $g(u)=|u|^{p-2}u$ with $p\in (2+\frac{8}{N},2^{*})$, $2^{*}=\frac{2N}{N-2}$ when $N\ge 3$. This problem is a nonlocal one as the appearance of the term $\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}$ implies that (1.1) is no longer a pointwise identity, which causes some mathematical difficulties and particularly interesting study.

Equation (1.1) is first proposed by Kirchhoff [1] in 1883 which can be seen as an extension of a class of the classical D’Alembert’s wave equation. It shows the free vibration of elastic strings and describes the motions of moderately large amplitude. For more physical and mathematical background on Kirchhoff type equations, we refer the readers to [2,3,4,5,6] and the references therein. Kirchhoff type problem has been intensively studied during the last decade since Lions [7] proposed an abstract functional analysis framework.

When looking for solutions to Eq. (1.1), there are two different ways according to the role of $\lambda$.

• (i) The frequency $\lambda$ is a fixed and assigned parameter;

• (ii) The frequency $\lambda$ is an unknown of the problem.

For case (i), to search for the solution to (1.1) equals to find a critical point of the action functional, defined on $H^{1}({\mathbb {R}}^{N})$ by

$$\begin{aligned} {\mathcal {A}}(u)=\frac{a}{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\lambda \Vert u\Vert ^{2}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x-\int _{{\mathbb {R}}^{N}}G(u)\textrm{d}x \end{aligned}$$

(1.3)

where $G(s):=\int _0^s g(t)\textrm{d}t$. Alternatively, one can search for solutions to (1.1) having prescribed mass; in this direction, $\lambda \in {\mathbb {R}}$ is part of the unknown and will appear as a Lagrange multiplier.

Normalized solutions of (1.1) are obtained by looking for critical points of the energy functional J(u) constrained on S(c), where

$$\begin{aligned} J(u)=\frac{a}{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x-\int _{{\mathbb {R}}^{N}}G(u)\textrm{d}x \end{aligned}$$

(1.4)

and

$$\begin{aligned} S(c)=\{u\in H^{1}({\mathbb {R}}^{N}):\Vert u\Vert ^{2}_{2}=c\}. \end{aligned}$$

(1.5)

It is standard to check that the above energy functional is of $C^{1}$. This case seems particularly meaningful from the physical point of view; here, we focus on this issue.

Recall that, in the case of non-potential case

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u - \lambda u =g(u). \end{aligned}$$

(1.6)

When $g(u)=|u|^{p-2}u$, Ye proposed that the mass critical exponent for Kirchhoff constraint minimization problem should be

$$\begin{aligned} {\overline{p}}=2+\frac{8}{N}. \end{aligned}$$

Theorem 1.1 in [8] shows that when $2<p<2+\frac{8}{N}$ in (1.6) which is called the mass subcritical case, energy functional is bounded from below on S(c); thus, a ground state as global minimizer for some $c>0$ can be found. When $p=2+\frac{8}{N}$, which is called mass critical case, the problem has no minimizer for all $c>0$. In the purely mass super-critical case, energy functional is unbounded from below, while Ye took a minimum on a suitable submanifold and proved that (1.6) has a couple of weak solution for any $c>0$ when $2+\frac{8}{N}<p<2^{*}$. Afterward, Luo and Wang [9] generalized the main results in [10, 11], obtaining the multiplicity of solutions in the dimension $N=3$ when $\frac{14}{3}<p<6$. Later in [12], Ye studied the concentration behavior of critical points with a minimax characterization when $p=2+\frac{8}{N}$, and got a mountain pass critical point for the functional on S(c). By some simple energy estimates, Zeng and Zhang [13] improved the results of [8], avoided using the concentration-compactness principles.

When g(u) is a general nonlinearity, we shall focus on the mass super-critical case. Chen and Xie [14] made the first progress in this direction by proving the existence and multiplicity results of solutions via minimizing method on a suitable submanifold. Tang and chen [15] extended and complemented the corresponding existence results in [8] in the presence of the variable potential K(x)g(u). For the more general nonlinear case, by using a detailed analysis via the blow up method, He et al., [16] proved the existence of ground state normalized solutions for any given $c>0$ and made clear the asymptotic behavior of these solutions as $c\rightarrow 0^{+}$ as well as $c\rightarrow +\infty$. Zhong and Zhang in [17] presented a novel global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions later on.

Next we briefly review the history of the potential case. In the purely mass subcritical case, Guo and Zhou [18] studied a Kirchhoff type elliptic equation with trapping potential when $1\le N\le 4$, where the existence and blow-up behavior of solutions with normalized $L^{2}$-norm for (1.1) with $g(u)=\beta |u|^{p-2}u$ were discussed. Later, Li and Hao [19] obtained the existence and nonexistence of energy minimizer in the dimension $N=4$ in the subcritical and critical exponent cases. By dealing with the minimization problem

$$\begin{aligned} e_{\beta }(b)=\inf \limits _{\left\{ u\in H^{1}({\mathbb {R}}^{2}):\int _{{\mathbb {R}}^{2}}u^{2}\textrm{d}x=1\right\} }E^{b}_{\beta }(u), \end{aligned}$$

where $E^{b}_{\beta }(u)$ is the energy functional, Zeng and Meng [20] proved the existence and asymptotic behavior of minimizers for the Kirchhoff functional with periodic potentials and $g(u)=|u|^{2}u$ in the dimension $N=2$. Zhu and Wang [21] studied the existence and nonexistence of ground states for the nonlinear Kirchhoff-Schrödinger equation with combined power nonlinearities. To the best of our knowledge, above results are all the case of trapping potential. This paper mainly studies the non-trapping potentials case, assume that $\limsup \limits _{|x|\rightarrow +\infty }V(x)=:V_{\infty }<+\infty$ exists. If $V_{\infty }\ne 0$, we can transform searching the weak normalized solution of (1.1) into the solution of the following equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u +(V(x)-V_{\infty })u+(\lambda +V_{\infty })u=g(u). \end{aligned}$$

The weak solution $(\lambda _c,u_c)$ of the above equation means that $(\lambda _c-V_{\infty },u_c)$ is the weak normalized solution of (1.1). For the sake of simplicity, we may suppose that $V_{\infty }=0$.

We firstly suppose that

$(V_{1})$ $\lim \limits _{|x|\rightarrow +\infty }V(x)=\sup \limits _{x\in {R}^{N}}V(x)=0$, $V(x)=V(|x|)$, there exists $\sigma _{1}\in \left[ 0,a-\frac{4a}{N(p-2)}\right)$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}|V(x)u^{2}\textrm{d}x\right| \le \sigma _{1}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{N}). \end{aligned}$$

$(V_{2})$ $\ \nabla V(x)$ exists for a.e. $x\in {\mathbb {R}}^{N}$, define

$$\begin{aligned} W(x) :=\frac{1}{2}\langle \nabla V(x),x\rangle . \end{aligned}$$

There exists $0\le \sigma _{2}<\min \{\frac{N(a-\sigma _{1})(p-2)}{4}-a,a\left( 1-\frac{N(p-2)}{2p}\right) \}$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}|W(x)u^{2}\textrm{d}x\right| \le \sigma _{2}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{N}). \end{aligned}$$

$(V_{3})$ $\ \nabla W(x)$ exists for a.e $x\in {\mathbb {R}}^{N}$, define

$$\begin{aligned} Y(x) :=\left( \frac{N(p-2)}{2}\right) W(x)+\langle \nabla W(x),x\rangle . \end{aligned}$$

There exists $\sigma _{3}\in \left[ 0,\frac{N(p-2)}{2}a-2a\right)$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}|Y(x)u^{2}\textrm{d}x\right| \le \sigma _{3}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{N}). \end{aligned}$$

Our main results read as follows.

Theorem 1.1

Let $N \ge 3$. Assume that $0\not \equiv V(x)$ satisfies $(V1)-(V3)$, then the Kirchhoff problem

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u + V(x)u +\lambda u=|u|^{p-2}u, \ \ {in} \ {{\mathbb {R}}^{N}}\\&\int _{{\mathbb {R}}^{N}}u^{2}\textrm{d}x=c. \end{aligned}\right. \end{aligned}$$

has a ground state normalized solution $(\lambda _{c},u_{c})\in ({\mathbb {R}}^{+},H^{1}_{r}({\mathbb {R}}^{N}))$.

We point out that Theorem 1.1 deals with the non-trapping potentials case when $g(u)=|u|^{p-2}u$, and it seems that no literatures involve the case of general mass super-critical nonlinearities with non-trapping potentials. Motivated by the research made in the mass super-critical case for the Schrödinger equation

$$\begin{aligned} -\Delta u+V(x)+\lambda u=g(u), \end{aligned}$$

the mass super-critical case with general nonlinearities and negative non-trapping potential will also be considered in present paper. To study the following results, we suppose further that

$(G_{1})$ $g\in C^{1}({\mathbb {R}},{\mathbb {R}})$ is odd.

$(G_{2})$ There exist $\alpha , \beta \in {\mathbb {R}}$ satisfying $\frac{14}{3}< \alpha \le \beta < 6$ such that

$$\begin{aligned} \alpha G(s)\le g(s)s \le \beta G(s) \ with \ G(s)=\int _0^s g(t)t \textrm{d}t. \end{aligned}$$

$(G_{3})$ The functional defined by ${\tilde{G}}:=\frac{1}{2}g(s)s-G(s)$ satisfies

$$\begin{aligned} {\tilde{G}}^{\prime }(s)s\ge \alpha {\tilde{G}}(s),\forall s\in {\mathbb {R}}. \end{aligned}$$

$(V_{4})$ $\lim \limits _{|x|\rightarrow +\infty }V(x)=\sup \limits _{x\in {R}^{N}}V(x)=0$, $V(x)=V(|x|)$, there exists $\sigma _{4}\in \left[ 0,a-\frac{4a}{3(\alpha -2)}\right)$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}|V(x)u^{2}\textrm{d}x\right| \le \sigma _{4}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{N}). \end{aligned}$$

$(V_{5})$ $\ \nabla V(x)$ exists for a.e. $x\in {\mathbb {R}}^{3}$, define

$$\begin{aligned} W(x) :=\frac{1}{2}\langle \nabla V(x),x\rangle . \end{aligned}$$

There exists $0\le \sigma _{5}<\min \{\frac{3(a-\sigma _{4})(\alpha -2)}{4}-a,a(\frac{3}{\beta }-\frac{1}{2})\}$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}|W(x)u^{2}\textrm{d}x\right| \le \sigma _{5}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{3}). \end{aligned}$$

$(V_{6})$ $\ \nabla W(x)$ exists for a.e. $x\in {\mathbb {R}}^{3}$, define

$$\begin{aligned} Z(x) :=(\frac{3}{2}\alpha -3)W(x)+\langle \nabla W(x),x\rangle . \end{aligned}$$

There exists $\sigma _{6}\in [0,\frac{3}{2}\alpha a-5a)$ such that

$$\begin{aligned} \left| \int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x\right| \le \sigma _{6}\Vert u\Vert ^{2}_{2}, \ \ \forall u \in H^{1}({\mathbb {R}}^{3}). \end{aligned}$$

Theorem 1.2

Let $N=3$, assume that $(G1)-(G3) \ and \ (V4)-(V6)$ hold, then Kirchhoff problem

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u + V(x)u +\lambda u=g(u), \ \ {in} \ {{\mathbb {R}}^{N}}\\&\int _{{\mathbb {R}}^{N}}u^{2}\textrm{d}x=c. \end{aligned}\right. \end{aligned}$$

has a ground state normalized solution $(\lambda _{c},u_{c})\in ({\mathbb {R}}^{+},H^{1}_{r}({\mathbb {R}}^{N}))$.

This paper is organized as follows. In Sect. 2, we collect some preliminary results which will be used in the rest the paper. Section 3 is devoted to the proof of Theorem 1.1. The proof of Theorem 1.2 is given in Sect. 4. In this paper, $H^{1}_{r}({\mathbb {R}}^{N})$ denotes the subspace of functions in $H^{1}({\mathbb {R}}^{N})$ which are radially symmetric with respect to 0; $\Vert u\Vert _{p}$ denotes the $L^{p}$-norm; $\rightharpoonup$ denotes weak convergence in $H^{1}({\mathbb {R}}^{N})$; Capital letters $C_{1},C_{2},\cdots$ denote positive constants which may depend on N and p, but never on u.

2 Preliminaries and Functional Setting

Let

$$\begin{aligned} {\mathcal {H}}:=\{u\in H^{1}({\mathbb {R}}^{N}):\left| \int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x\right| <+\infty \} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u\Vert _{{\mathcal {H}}}:=\left( \int _{{\mathbb {R}}^{N}}|\nabla u|^{2}+V(x)u^{2}+u^{2}\textrm{d}x\right) ^{\frac{1}{2}}, \end{aligned}$$

which is equivalent to the usual norm $\Vert u\Vert _{H^{1}({\mathbb {R}}^{N})}$ under our assumption (V1) or (V4) in present paper. To find normalized solution, we define the normalization constraint

$$\begin{aligned} S_{c}:=\{u\in {{\mathcal {H}}}:\int _{{\mathbb {R}}^{N}}u^{2}\textrm{d}x=c\}. \end{aligned}$$

For $u\in H^{1}({\mathbb {R}}^{N}), t\in {\mathbb {R}}^{+},$ we consider the fiber map: $u(x)\mapsto (t\star u): =t^{\frac{N}{2}}u(tx)$, which preserves $L^{2}$-norm. Hence, it is natural to define the maps

$$\begin{aligned}{} & {} (\Psi _{u})(t):=J[t\star u]=\frac{a}{2}t^{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(t^{-1}x)u^{2}\textrm{d}x\\{} & {} \quad -t^{-N}\int _{{\mathbb {R}}^{N}}G(t^{\frac{N}{2}}u)\textrm{d}x \\{} & {} \quad (\Psi _{\infty ,u})(t):=I[t\star u]=\frac{a}{2}t^{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}-t^{-N}\int _{{\mathbb {R}}^{N}}G(t^{\frac{N}{2}}u)\textrm{d}x, \end{aligned}$$

where

$$\begin{aligned} I[u]=\frac{a}{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{N}}G(u)\textrm{d}x. \end{aligned}$$

Direct calculation gives

$$\begin{aligned} (\Psi _{u})^{\prime }(t)&=at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-\frac{1}{2}\int _{{\mathbb {R}}^{N}}\langle \nabla V(t^{-1}x),t^{-2}x\rangle u^{2}\textrm{d}x\nonumber \\&-Nt^{-N-1}\int _{{\mathbb {R}}^{N}}\left[ \frac{1}{2}g(t^{\frac{N}{2}}u)t^{\frac{N}{2}}u-G(t^{\frac{N}{2}}u)\right] \textrm{d}x \nonumber \\&=\frac{1}{t}P[t\star u] \end{aligned}$$

(2.1)

and

$$\begin{aligned} (\Psi _{\infty ,u})^{\prime }(t)&=at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-Nt^{-N-1}\int _{{\mathbb {R}}^{N}}\left[ \frac{1}{2}g(t^{\frac{N}{2}}u)t^{\frac{N}{2}}u-G(t^{\frac{N}{2}}u)\right] \textrm{d}x \\&=\frac{1}{t}P_{\infty }[t\star u], \end{aligned}$$

where

$$\begin{aligned} P[u]=a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}-\frac{1}{2}\int _{{\mathbb {R}}^{N}}\langle \nabla V(x),x\rangle u^{2}\textrm{d}x\!-\!N\!\int _{{\mathbb {R}}^{N}}\left[ \frac{1}{2}g(u)u-G(u)\right] \!\textrm{d}x \end{aligned}$$

and

$$\begin{aligned} P_{\infty }[u]=a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}-N\int _{{\mathbb {R}}^{N}}\left[ \frac{1}{2}g(u)u-G(u)\right] \textrm{d}x. \end{aligned}$$

Define the Pohozaev manifold

$$\begin{aligned} {\mathcal {P}}:=\left\{ u\in H^{1}({\mathbb {R}}^{N}):P[u]=0\right\} \ \ {and}\ \ {\mathcal {P}}_{\infty }:=\left\{ u\in H^{1}({\mathbb {R}}^{N}):P_{\infty }[u]=0\right\} . \end{aligned}$$

For $c>0$, set

$$\begin{aligned}{} & {} {\mathcal {P}}_{c}:={S_{c}\cap {\mathcal {P}}}, \ {\mathcal {P}}_{\infty ,c}:={S_{c}\cap {\mathcal {P}}_{\infty }}\\{} & {} \quad {\mathcal {P}}^{r}_{c}:={S_{c}\cap {\mathcal {P}}\cap H^{1}_{r}({\mathbb {R}}^{N})}, \ {\mathcal {P}}^{r}_{\infty ,c}:={S_{c}\cap {\mathcal {P}}_{\infty }\cap H^{1}_{r}({\mathbb {R}}^{N})}. \end{aligned}$$

Then, we have the following lemmas.

Lemma 2.1

Suppose that $u \in H^{1}({\mathbb {R}}^{N})$ is the solution to (1.1), then $u\in {\mathcal {P}}.$

Proof

Let u be a test function, then we obtain

$$\begin{aligned} a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}+\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x+\lambda \Vert u\Vert ^{2}_{2}=\int _{{\mathbb {R}}^{N}}g(u)u\textrm{d}x. \end{aligned}$$

(2.2)

Moreover, u satisfies the following Pohozaev identity

$$\begin{aligned}{} & {} (N-2)\left( a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}\right) +\int _{{\mathbb {R}}^{N}}\langle \nabla V(x),x\rangle u^{2}\textrm{d}x \nonumber \\{} & {} \quad +N\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x+N\lambda \Vert u\Vert ^{2}_{2}-2N\int _{{\mathbb {R}}^{N}}G(u)\textrm{d}x=0. \end{aligned}$$

(2.3)

Hence, we have that

$$\begin{aligned} a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}-\frac{1}{2}\int _{{\mathbb {R}}^{N}}\langle \nabla V(x),x\rangle u^{2}\textrm{d}x-N\int _{{\mathbb {R}}^{N}}\left[ \frac{1}{2}g(u)u-G(u)\right] \textrm{d}x=0. \end{aligned}$$

$\square$

Proposition 2.2

Let $u\in S_{c}$ and $t\in {\mathbb {R}}^{+}$. Then, t is a critical point for $(\Psi _{u})(t)$ if and only if $t\star u\in {\mathcal {P}}_{c}.$

Proof

The simple proof follows from the fact that $(\Psi _{u})^{\prime }(t)=\frac{1}{t}P[t\star u].$

Lemma 2.3

Assume that $\Psi _{u}^{\prime \prime }(1)\ne 0$, then ${\mathcal {P}}_{c}$ is a natural constraint of $J\mid _{S_{c}}$.

Proof

Let u be a critical point for $J\mid _{{\mathcal {P}}_{c}},$ that is, there exists $(\lambda ,\mu )\in ({\mathbb {R}},{\mathbb {R}})$ such that

$$\begin{aligned} J^{\prime }[u]+\lambda u+\mu P^{\prime }[u]=0. \end{aligned}$$

(2.4)

It is enough to show that $\mu =0$. Indeed, since u is a solution of (2.4), it satisfies the following Pohozaev identity

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\Phi [t\star u]\bigg |_{t=1}=0, \end{aligned}$$

where $\Phi [u]=J[u]+\frac{1}{2}\lambda \Vert u\Vert ^{2}_{2}+\mu P[u].$ Then,

$$\begin{aligned} \Phi [t\star u]&=J[t\star u]+\frac{1}{2}\lambda \Vert u\Vert ^{2}_{2}+\mu P[t\star u]\\&=\Psi _{u}(t)+\frac{1}{2}\lambda \Vert u\Vert ^{2}_{2}+\frac{\mu }{t}(\Psi _{u}){'}(t). \end{aligned}$$

And hence,

$$\begin{aligned} 0=\frac{\textrm{d}}{\textrm{d}t} \Phi [t\star u]\bigg |_{t=1}&=(1-\mu )(\Psi _{u}){'}(1)+\mu (\Psi _{u}){''}(1)\\&=(1-\mu )P[u]+\mu (\Psi _{u}){''}(1)\\&=\mu (\Psi _{u}){''}(1). \end{aligned}$$

$\square$

3 Normalized Solution: The Mass Super-critical Case for $N\ge 3$

In this section, we shall prove that for $2+\frac{8}{N}<p<\frac{2N}{N-2}$, Theorem 1.1 holds for any $a,b>0,V(x)$ satisfying $(V1)-(V3)$. We fist show that the existence of bounded minimizing sequence, then deal with the convergence.

Lemma 3.1

For $a,b>0$, the following Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u +\lambda u=|u|^{p-2}u \end{aligned}$$

(3.1)

has no nontrivial solution $u\in H^{1}({\mathbb {R}}^{N})$ provided $\lambda \le 0$.

Proof

Seeking for a contradiction, let we assume that there exists a $0\not \equiv u\in H^{1}({\mathbb {R}}^{N})$ solving (3.1) with $\lambda \le 0$, then

$$\begin{aligned} a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}+\lambda \Vert u\Vert ^{2}_{2}=\Vert u\Vert ^{p}_{p}. \end{aligned}$$

(3.2)

Moreover, u satisfies the following Pohozaev identity

$$\begin{aligned} (N-2)(a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2})+N\lambda \Vert u\Vert ^{2}_{2}-\frac{2N}{p}\Vert u\Vert ^{p}_{p}=0. \end{aligned}$$

(3.3)

The equality (3.2) and (3.3) imply that

$$\begin{aligned} \lambda \Vert u\Vert ^{2}_{2}=\left( 1-\frac{N(p-2)}{2p}\right) \Vert u\Vert ^{p}_{p}. \end{aligned}$$

Thus if $\lambda <0$, then we obtain that

$$\begin{aligned} 0>\lambda \Vert u\Vert ^{2}_{2}=\left( 1-\frac{N(p-2)}{2p}\right) \Vert u\Vert ^{p}_{p}\ge 0, \end{aligned}$$

which is a contradiction. On the other hand, if $\lambda =0$, then there holds $\Vert u\Vert ^{p}_{p}=0$. By (3.2) we have that $\Vert \nabla u\Vert ^{2}_{2}=0$, a contradiction to $0\not \equiv u\in H^{1}({\mathbb {R}}^{N}).$ $\square$

Lemma 3.2

Under the assumption (V2), there exists $\delta _{c}>0$ such that

$$\begin{aligned} \inf \left\{ t>0:\exists u\in S_{c} \ with \ \Vert \nabla u\Vert ^{2}_{2}=1, \ such \ that \ t\star u\in {\mathcal {P}}_{c}\right\} \ge \delta _{c}, \end{aligned}$$

(3.4)

for any $c>0$. Therefore,

$$\begin{aligned} \inf \limits _{u\in {\mathcal {P}}_{c}}\Vert \nabla u\Vert ^{2}_{2}\ge \delta ^{2}_{c}. \end{aligned}$$

(3.5)

Proof

By a straightforward calculation, it follows that

$$\begin{aligned} (\Psi _{u})^{\prime }(t)= & {} at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x\\{} & {} -\frac{N(p-2)}{2p}t^{\frac{N(p-2)}{2}-1}\Vert u\Vert ^{p}_{p}. \end{aligned}$$

From (V2),

$$\begin{aligned} at\Vert \nabla u\Vert ^{2}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x&=at\Vert \nabla u\Vert ^{2}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W(x)(t\star u)^{2}\textrm{d}x\\&\ge (a-\sigma _{2})\Vert \nabla u\Vert ^{2}_{2}t. \end{aligned}$$

For any $u\in S_{c}$ with $\Vert \nabla u\Vert ^{2}_{2}=1$ such that $t\star u\in {\mathcal {P}}_{c}$, by $Propositon \ 2.2$, we can obtain $(\Psi _{u})^{\prime }(t)=0$, then

$$\begin{aligned} a-\sigma _{2}&=(a-\sigma _{2})\Vert \nabla u\Vert ^{2}_{2} \\&\le a\Vert \nabla u\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x t^{-2}\\&\le a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}t^{2}-\int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x t^{-2}\\&=\frac{N(p-2)}{2p}t^{\frac{N(p-2)}{2}-2}\Vert u\Vert ^{p}_{p}. \end{aligned}$$

Now $p>2+\frac{8}{N}$ and (V2) imply that there exists a lower bound $\delta _{c}>0$. Thus, $\inf \limits _{u\in {\mathcal {P}}_{c}}\Vert \nabla u\Vert ^{2}_{2}\ge \delta ^{2}_{c}.$ $\square$

Notice the well properties of $\Psi _{u}(t)$, to determine the exact type and location of critical point for $J\mid _{S_{c}}$, we consider the decomposition of ${\mathcal {P}}_{c}$ into the disjoint union ${\mathcal {P}}_{c}={\mathcal {P}}^{+}_{c} \ \bigcup \ {\mathcal {P}}^{0}_{c} \ \bigcup \ {\mathcal {P}}^{-}_{c},$ where

$$\begin{aligned}{} & {} {\mathcal {P}}^{+}_{c}:=\left\{ u\in {\mathcal {P}}_{c}:(\Psi _{u})^{\prime \prime }(1)>0\right\} =\left\{ u\in S_{c}:(\Psi _{u})^{\prime }(1)=0,(\Psi _{u})^{\prime \prime }(1)>0\right\} \nonumber \\{} & {} {\mathcal {P}}^{0}_{c}:=\left\{ u\in {\mathcal {P}}_{c}:(\Psi _{u})^{\prime \prime }(1)=0\right\} =\left\{ u\in S_{c}:(\Psi _{u})^{\prime }(1)=0,(\Psi _{u})^{\prime \prime }(1)=0\right\} \nonumber \\{} & {} {\mathcal {P}}^{-}_{c}:=\left\{ u\in {\mathcal {P}}_{c}:(\Psi _{u})^{\prime \prime }(1)<0\right\} =\left\{ u\in S_{c}:(\Psi _{u})^{\prime }(1)=0,(\Psi _{u})^{\prime \prime }(1)<0\right\} .\qquad \quad \end{aligned}$$

(3.6)

Lemma 3.3

Assume (V3) holds, then ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$ is closed in $H^{1}({\mathbb {R}}^{N})$.

Proof

For any $u\in {\mathcal {P}}$, we have

$$\begin{aligned} P[u]=a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x-\frac{N(p-2)}{2p}\Vert u\Vert ^{p}_{p}=0. \end{aligned}$$

Notice that

$$\begin{aligned} (\Psi _{u})^{\prime }(t)= & {} at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x\\{} & {} -\frac{N(p-2)}{2p}t^{\frac{N(p-2)}{2}-1}\Vert u\Vert ^{p}_{p}, \end{aligned}$$

together with (V3) yields immediately

$$\begin{aligned}&(\Psi _{u})^{\prime \prime }(1)\\&\quad =a\Vert \nabla u\Vert ^{2}_{2}+3b\Vert \nabla u\Vert ^{4}_{2}\\&\qquad +\int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x+\int _{{\mathbb {R}}^{N}}\langle \nabla W(x),x\rangle u^{2}\textrm{d}x-\frac{N(p-2)}{2p}\left( \frac{N(p-2)}{2}-1\right) \Vert u\Vert ^{p}_{p}\\&\quad =a\left( 2-\frac{N(p-2)}{2}\right) \Vert \nabla u\Vert ^{2}_{2}+b\left( 4-\frac{N(p-2)}{2}\right) \Vert \nabla u\Vert ^{4}_{2}\\&\qquad +\frac{N(p-2)}{2}\int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x +\int _{{\mathbb {R}}^{N}}\langle \nabla W(x),x\rangle u^{2}\textrm{d}x \\&\quad \le a\left( 2-\frac{N(p-2)}{2}\right) \Vert \nabla u\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{N}}Y(x)u^{2}\textrm{d}x\\&\quad \le \left[ a\left( 2-\frac{N(p-2)}{2}\right) +\sigma _{3}\right] \Vert \nabla u\Vert ^{2}_{2}<0. \end{aligned}$$

The last inequality means ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$, which implies that ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$ is closed in $H^{1}({\mathbb {R}}^{N})$. $\square$

Remark 3.4

From Lemma 2.3 and lemma 3.2, we know $J|_{{\mathcal {P}}^{-}_{c}}$ is a natural constraint of $J|_{S_{c}}.$

Remark 3.5

Assume that $\left\{ u_{n}\right\} \subset {\mathcal {P}}^{-}_{c}$ such that $J[u_{n}]$ approaches a possible critical value. Since the constraint$(\Psi _{u_{n}})^{\prime \prime }(1)<0$ is open, by the Lagrange multipliers rule, there exist $\lambda _{n},\mu _{n}\in {\mathbb {R}}$ such that

$$\begin{aligned} J^{\prime }[u_{n}]+\lambda _{n}u_{n}+\mu _{n}P^{\prime }[u_{n}]\rightarrow 0. \end{aligned}$$

Then by similar proof as Lemma 2.3, we see that

$$\begin{aligned} \mu _{n}(\Psi _{u_{n}})^{\prime \prime }(1)\rightarrow 0. \end{aligned}$$

Notice that $(\Psi _{u_{n}})^{\prime \prime }(1)<0$, so

$$\begin{aligned} \mu _{n}\Vert \nabla u_{n}\Vert ^{2}_{2}\rightarrow 0. \end{aligned}$$

(3.7)

On the other hand, by Lemma 3.2, we obtain that $\Vert \nabla u\Vert ^{2}_{2}\ge \delta _{c}^{2}>0$ for all $u\in {\mathcal {P}}_{c}$. Thus, we have that $\mu _{n}\rightarrow 0$. Therefore, if ${u_{n}}$ is bounded in $H^{1}({\mathbb {R}}^{N})$, then we can deduce that

$$\begin{aligned} J^{\prime }[u_{n}]+\lambda _{n}u_{n}\rightarrow 0 \ in \ H^{-1}({\mathbb {R}}^{N}). \end{aligned}$$

Corollary 3.6

Let the assumptions (V1),(V2) hold, for any $u\in H^{1}({\mathbb {R}}^{N})\setminus \left\{ 0\right\}$, there exists a unique $t_{u}>0$ such that $t_{u}\star u\in {\mathcal {P}}$. Moreover,

$$\begin{aligned} J[t_{u}\star u]=\max \limits _{t>0}J[t\star u]. \end{aligned}$$

(3.8)

Proof

We first show the existence of $t_{u}$. For any $u\in H^{1}({\mathbb {R}}^{N})\setminus \left\{ 0\right\}$, we have that $\Vert \nabla u\Vert _{2}>0$. Let ${\tilde{m}}:=\Vert u\Vert ^{2}_{2}$, by (V2), we obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}xt^{-1}&=\int _{{\mathbb {R}}^{N}}W(x)(t\star u)^{2}\textrm{d}xt^{-1} \\&\le \sigma _{2}\Vert \nabla (t\star u)\Vert ^{2}_{2}t^{-1}\\&=\sigma _{2}t\Vert \nabla u\Vert ^{2}_{2}. \end{aligned}$$

Then,

$$\begin{aligned} (\Psi _{u})^{\prime }(t)&=a\Vert \nabla u\Vert ^{2}_{2}t+b\Vert \nabla u\Vert ^{4}_{2}t^{3}-\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}xt^{-1}\\&\quad -\frac{N(p-2)}{2p}\Vert u\Vert ^{p}_{p}t^{\frac{N(p-2)}{2}-1}\\&\ge (a-\sigma _{2})\Vert \nabla u\Vert ^{2}_{2}t+b\Vert \nabla u\Vert ^{4}_{2}t^{3}-\frac{N(p-2)}{2p}\Vert u\Vert ^{p}_{p}t^{\frac{N(p-2)}{2}-1}. \end{aligned}$$

one can see that

$$\begin{aligned} (\Psi _{u})^{\prime }(t)>0, \ t\rightarrow 0. \end{aligned}$$

(3.9)

Therefore, there exists $t_{0}>0$ such that $\Psi _{u}(t)$ is increasing in $t \in (0,t_{0})$. On the other hand, by (V1), we have that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}V(x)(t\star u)^{2}\le \sigma _{1}\Vert \nabla (t\star u)\Vert ^{2}_{2}=\sigma _{1}t^{2}\Vert \nabla u\Vert ^{2}_{2}. \end{aligned}$$

Then,

$$\begin{aligned} \Psi _{u}(t)&=J[t\star u]\\&=\frac{a}{2}t^{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)(t \star u)^{2}\textrm{d}x-\frac{1}{p}t^{\frac{N(p-2)}{2}}\Vert u\Vert ^{p}_{p}\\&\le \frac{t^{2}}{2}(a+\sigma _{1})\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}-\frac{1}{p}t^{\frac{N(p-2)}{2}}\Vert u\Vert ^{p}_{p}. \end{aligned}$$

Hence,

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }\Psi _{u}(t)=\lim \limits _{t\rightarrow +\infty }J[t\star u]=-\infty . \end{aligned}$$

(3.10)

By (3.9) (3.10), we deduce that there exists $t_{1}>t_{0}$ such that

$$\begin{aligned} J[t_{1}\star u]=\max \limits _{t>0}J[t\star u]. \end{aligned}$$

(3.11)

Consequently $(\Psi _{u})^{\prime }(t_{1})=0$ and Proposition 2.2 implies that $t_{1}\star u\in {\mathcal {P}}$.

We next to show the uniqueness of $t_{u}$. Assume that there exists $t_{2}>0$ such that $t_{2}\star u\in {\mathcal {P}}$. It follows Lemma 3.3 that both $t_{1}$ and $t_{2}$ are strict local maximum of $\Psi _{u}(t)$. Without loss of generality, we suppose that $t_{1}<t_{2}$, then there exists $t_{3}\in (t_{1},t_{2})$ such that

$$\begin{aligned} \Psi _{u}(t_{3})=\min \limits _{t\in [t_{1},t_{2}]}\Psi _{u}(t). \end{aligned}$$

Hence, we obtain that $(\Psi _{u})^{\prime }(t_{3})=0$, $(\Psi _{u})^{\prime \prime }(t_{3})\ge 0$, which implies that $t_{3}\star u\in {\mathcal {P}}^{+}_{c} \ \bigcup \ {\mathcal {P}}^{0}_{c}$. This leads to a contradiction by Lemma 3.3. $\square$

Remark 3.7

Under (V1)-(V3), for any $u\in {\mathcal {P}}_{c}$, we have that $\Psi _{u}(t)\rightarrow 0 \ as \ t\rightarrow 0^{+}$ and $\Psi _{u}(t)\rightarrow -\infty \ as \ t\rightarrow +\infty$. Corollary 3.6 implies that

$$\begin{aligned} J[u]=\max \limits _{t>0}J[t\star u]>0. \end{aligned}$$

Define

$$\begin{aligned} M_{c}:=\inf \limits _{u\in {\mathcal {P}}_{c}}J[u]=\inf \limits _{u\in S_{c}}\max \limits _{t>0}J[t\star u], \ m_c:=\inf \limits _{u\in {\mathcal {P}}_{\infty ,c}}I[u]=\inf \limits _{u\in S_{c}}\max \limits _{t>0}I[t\star u].\nonumber \\ \end{aligned}$$

(3.12)

Lemma 3.8

Under the hypotheses (V1)–(V3), $M_{c}>0$.

Proof

By (V2), we have

$$\begin{aligned} (a+\sigma _{2})\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}&\ge a\Vert \nabla u\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x+b\Vert \nabla u\Vert ^{4}_{2}\\&=\frac{N(p-2)}{2p}\Vert u\Vert ^{p}_{p}. \end{aligned}$$

Then,

$$\begin{aligned} J[u]&=\frac{a}{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x-\frac{1}{p}\Vert u\Vert ^{p}_{p} \\&\ge \frac{1}{2}(a-\sigma _{1})\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}-\frac{1}{p}\Vert u\Vert ^{p}_{p}\\&\ge \left[ \frac{1}{2}(a-\sigma _{1})-\frac{2(a+\sigma _{2})}{N(p-2)}\right] \Vert \nabla u\Vert ^{2}_{2}+\left[ \frac{b}{4}-\frac{2b}{N(p-2)}\right] \Vert \nabla u\Vert ^{4}_{2}. \end{aligned}$$

Since $p>2+\frac{8}{N}$ and (V2), we deduce that $M_{c}>0$. $\square$

Corollary 3.9

Under the hypotheses (V1)-(V3), we have that

$$\begin{aligned} \lim \limits _{u\in {\mathcal {P}}_{c}, \Vert \nabla u\Vert ^{2}_{2}\rightarrow \infty }J[u]=+\infty . \end{aligned}$$

(3.13)

Proof

By Lemma 3.8, we know

$$\begin{aligned} J[u]\ge \left[ \frac{1}{2}(a-\sigma _{1})-\frac{2(a+\sigma _{2})}{N(p-2)}\right] \Vert \nabla u\Vert ^{2}_{2}+\left[ \frac{b}{4}-\frac{2b}{N(p-2)}\right] \Vert \nabla u\Vert ^{4}_{2}, \end{aligned}$$

then the corollary follows. $\square$

By a series results from above lemmas, we can find a bounded minimizing sequence in $H^{1}({\mathbb {R}}^{N})$ which cannot be compact in any $L^{p}({\mathbb {R}}^{N})$. Due to the presence of the nonlocal term, we have to overcome some difficulties, which is different from the local problem, to gain the compactness of a $(PS)_c$ sequence. In order to complete the proof of Theorem 1.1, we will apply the principle of symmetric criticality, working in a radial setting $H^{1}_{r}({\mathbb {R}}^{N})$.

Proof of Theorem 1.1

Let $\{u_{n}\}\subset {\mathcal {P}}^{rad}_{c}$ be a minimizing sequence for J, namely $J[u_{n}]\rightarrow M_{c}$.

Claim $M_{c}<m_{c}$. Indeed, assume that the level $m_{c}$ is attained by $w_{c}\in S_{c}$, using Lemma 3.1, we get the corresponding Lagrange multiplier $\lambda _{\infty ,c}\ge 0$. Clearly, by the Brezis–Kato theorem, elliptic regularity theory and strong maximum principle (Detailed proof can be referred to [22] Lemma 1.30 and [23] Lemma 2.1), we deduce that $w_{c}(x)>0$ in ${\mathbb {R}}^{N}$. By (V1), one can see that

$$\begin{aligned}&M_{c}\le \max \limits _{t>0}J[t\star w_{c}]=J[t_{w_{c}}\star w_{c}]=I[t_{w_{c}}\star w_{c}]+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)w_{c}(x)^{2}\textrm{d}x\\&\quad <I[t_{w_{c}}\star w_{c}]\le \max \limits _{t>0}I[t\star w_{c}]=I[w_{c}]=m_{c}. \end{aligned}$$

By Corollary 3.9, we have that $\{u_{n}\}$ is a bounded minimizing sequence in $H^{1}_{r}({\mathbb {R}}^{N})$. Up to a subsequence, we may assume that $u_{n}\rightharpoonup u$ in $H^{1}_{r}({\mathbb {R}}^{N})$.

Next we show that $u\ne 0$. Otherwise, if $u=0$, then it is not hard to show that $I[u_{n}]=M_{c}+o(1)$ and $(\Psi _{\infty ,u_{n}})^{\prime }(1)=o(1)$. Hence, there exists $t_{n}=1+o(1)$ such that $t_{n}\star u_{n}\in {\mathcal {P}}_{\infty ,c}$. Then,

$$\begin{aligned} m_{c}\le I[t_{n}\star u_{n}]=I[u_{n}]+o(1)=M_{c}+o(1), \end{aligned}$$

which leads to a contradiction to $M_{c}<m_{c}$.

Since $\{u_{n}\}$ is bounded in $H^{1}_{r}({\mathbb {R}}^{N})$, one can see that

$$\begin{aligned} \lambda _{n}=-\frac{\langle J^{\prime }[u_{n}],u_{n}\rangle }{c} \end{aligned}$$

is bounded. Meanwhile

$$\begin{aligned}&\lambda _{n}\Vert u_{n}\Vert ^{2}_{2}=-\langle J^{\prime }[u_{n}],u_{n}\rangle \\&\quad =\Vert u_{n}\Vert ^{p}_{p}-a\Vert \nabla u_{n}\Vert ^{2}_{2}-b\Vert \nabla u_{n}\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{N}}V(x)u^{2}_{n}\textrm{d}x\\&\quad \ge \Vert u_{n}\Vert ^{p}_{p}-a\Vert \nabla u_{n}\Vert ^{2}_{2}-b\Vert \nabla u_{n}\Vert ^{4}_{2}\\&\quad =\Vert u_{n}\Vert ^{p}_{p}-\left( \int _{{\mathbb {R}}^{N}}W(x)u^{2}_{n}\textrm{d}x+\frac{N(p-2)}{2p}\Vert u_{n}\Vert ^{p}_{p}\right) \\&\quad \ge \left[ 1-\frac{N(p-2)}{2p}\right] \Vert u_{n}\Vert ^{p}_{p}-\sigma _{2}\Vert \nabla u_{n}\Vert ^{2}_{2}. \end{aligned}$$

On the other hand,

$$\begin{aligned}&(a-\sigma _{2})\Vert \nabla u_{n}\Vert ^{2}_{2}\le (a-\sigma _{2})\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2}\\&\quad \le a\Vert \nabla u_{n}\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{N}}W(x)u^{2}_{n}\textrm{d}x+b\Vert \nabla u_{n}\Vert ^{4}_{2}\\&\quad =\frac{N(p-2)}{2p}\Vert u_{n}\Vert ^{p}_{p}. \end{aligned}$$

Therefore,

$$\begin{aligned} \lambda _{n}\Vert u_{n}\Vert ^{2}_{2}\ge \left\{ \left[ 1-\frac{N(p-2)}{2p}\right] \frac{2p}{N(p-2)}(a-\sigma _{2})-\sigma _{2}\right\} \Vert \nabla u_{n}\Vert ^{2}_{2}. \end{aligned}$$

By (V2), we get

$$\begin{aligned} \left[ 1-\frac{N(p-2)}{2p}\right] \frac{2p}{N(p-2)}(a-\sigma _{2})-\sigma _{2}>0. \end{aligned}$$

Hence by Lemma 3.2, there exists $\delta >0$ such that $\lambda _{n}c>\delta$, $\forall n\in {\mathbb {N}}$. Since $\{\lambda _{n}\}$ is bounded, up to a subsequence, we may suppose there exists $\lambda >0$ such that $\lambda _{n}\rightarrow \lambda$.

Let assume $\Vert \nabla u_{n}\Vert ^{2}_{2}\rightarrow \theta$, then it is easy to see that u solves

$$\begin{aligned} -(a+b\theta )\Delta u+\lambda u+V(x)u=|u|^{p-2}u \ in \ {\mathbb {R}}^{N}. \end{aligned}$$

(3.14)

Hence, for all $u\in {\mathcal {P}}^{rad}_{c}$, we have

$$\begin{aligned}&a\Vert \nabla u\Vert ^{2}_{2}+b\theta \Vert \nabla u\Vert ^{2}_{2}=\frac{1}{2}\int _{{\mathbb {R}}^{N}}\langle \nabla V(x),x\rangle u^{2}\textrm{d}x+\frac{N(p-2)}{2p}\Vert u\Vert ^{p}_{p}\\&\quad =\lim \limits _{n\rightarrow \infty }\left( \frac{1}{2}\int _{{\mathbb {R}}^{N}}\langle \nabla V(x),x\rangle u_{n}^{2}\textrm{d}x+\frac{N(p-2)}{2p}\Vert u_{n}\Vert ^{p}_{p}\right) \\&\quad =\lim \limits _{n\rightarrow \infty }\left( a\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2}\right) \\&\quad =a\theta +b\theta ^{2}. \end{aligned}$$

So $(a+b\theta )(\Vert \nabla u\Vert ^{2}_{2}-\theta )=0$. By $a>0,b>0,\theta \ge 0$, we get that $\Vert \nabla u\Vert ^{2}_{2}=\theta$. Hence, (3.14) implies that

$$\begin{aligned} -(a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\textrm{d}x)\Delta u+\lambda u+V(x)u=|u|^{p-2}u \ in \ {\mathbb {R}}^{N}. \end{aligned}$$

Now we obtain

$$\begin{aligned}&a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}+\lambda \Vert u\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x\\&\quad =\Vert u\Vert ^{p}_{p} \\&\quad =\Vert u_{n}\Vert ^{p}_{p}+o(1)\\&\quad =a\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2}+\lambda \Vert u_{n}\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{N}}V(x)u^{2}_{n}\textrm{d}x+o(1), \end{aligned}$$

by which it follows that $\lambda (\Vert u_{n}\Vert ^{2}_{2}-\Vert u\Vert ^{2}_{2})=0$, so $\Vert u\Vert ^{2}_{2}=\Vert u_{n}\Vert ^{2}_{2}=c$. Thus, $u_{n}\rightarrow u \ in \ L^{2}_{r}({\mathbb {R}}^{N}),$ $u_{n}\rightarrow u \ in \ H^{1}_{r}({\mathbb {R}}^{N})$ and $u\in {\mathcal {P}}^{rad}_{c}$. On the other hand, since $\Vert u_{n}\Vert ^{p}_{p}\rightarrow \Vert u\Vert ^{p}_{p}$, we have

$$\begin{aligned} J[u]=\lim \limits _{n\rightarrow \infty }J[u_{n}]=M_{c}. \end{aligned}$$

By Lemma 2.3, there exists some $\lambda \in {\mathbb {R}}^{+}$ such that $(\lambda _{c},u_{c}):=(\lambda ,u)$ solves (1.1)–(1.2).$\square$

4 Normalized Solution: The Mass Super-critical Case for $N=3$

In this section, we assume $N=3$, g is a rather general nonlinearity satisfying certain assumptions. Similar but more complicated arguments work to show the existence of normalized solutions to the following nonlinear mass super-critical Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u + V(x)u +\lambda u=g(u),\ \ \ u\in H^{1}_{r}({\mathbb {R}}^{N}). \end{aligned}$$

Lemma 4.1

For $N=3,\ a,b>0$, suppose that there exist some $2<\alpha \le \beta <6$ such that

$$\begin{aligned} \alpha G(s)\le g(s)s\le \beta G(s),\ \ \ \forall s \in {\mathbb {R}}. \end{aligned}$$

(4.1)

Then, the following Kirchhoff equation

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^{N}}|\nabla u|^{2}\right) \triangle u +\lambda u=g(u) \end{aligned}$$

(4.2)

has no nontrivial solution $u\in H^{1}({\mathbb {R}}^{N})$ provided $\lambda \le 0$.

Proof

We argue by contradiction and assume that there exists some $0\not \equiv u\in H^{1}({\mathbb {R}}^{N})$ solving (4.2) with $\lambda \le 0$. By (4.2), we obtain

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

(4.3)

Moreover, u satisfies the following Pohozaev identity

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

(4.4)

Since $\lambda \le 0$, the equality (4.3) and (4.4) imply that

$$\begin{aligned} 0\ge \lambda \Vert u\Vert ^{2}_{2}=\int _{{\mathbb {R}}^{N}}\left[ 3G(u)-\frac{1}{2}g(u)u\right] \textrm{d}x. \end{aligned}$$

For $N=3$, by (4.1), we obtain that $\int _{{\mathbb {R}}^{3}}\left[ 3G(u)-\frac{1}{2}g(u)u\right] \textrm{d}x\ge 0$, which implies $\lambda =0$ and

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}G(u)\textrm{d}x=\int _{{\mathbb {R}}^{3}}g(u)u\textrm{d}x=\int _{{\mathbb {R}}^{3}}{\tilde{G}}(u)\textrm{d}x. \end{aligned}$$

However, for $\lambda =0$, noting that $u\in {\mathcal {P}}_{\infty }$, $P_{\infty }[u]=0$, we can also have $\Vert \nabla u\Vert _{2}=0$, which is a contradiction. $\square$

Lemma 4.2

Under the assumptions $(G1)-(G2) \ and \ (V5)$, there exists $\delta _{c}>0$ such that

$$\begin{aligned} \inf \left\{ t>0:\exists u\in S_{c} \ with \ \Vert \nabla u\Vert ^{2}_{2}=1, \ such \ that \ t\star u\in {\mathcal {P}}_{c}\right\} \ge \delta _{c}, \end{aligned}$$

(4.5)

for any $c>0$. Therefore,

$$\begin{aligned} \inf \limits _{u\in {\mathcal {P}}_{c}}\Vert \nabla u\Vert ^{2}_{2}\ge \delta ^{2}_{c}. \end{aligned}$$

(4.6)

Proof

Direct calculation gives

$$\begin{aligned} (\Psi _{u})^{\prime }(t)=at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x-3t^{-4}\int _{{\mathbb {R}}^{N}}{\tilde{G}}(t^{\frac{3}{2}}u)\textrm{d}x. \end{aligned}$$

By (V5),

$$\begin{aligned}&at\Vert \nabla u\Vert ^{2}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x=at\Vert \nabla u\Vert ^{2}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W(x)(t\star u)^{2}\textrm{d}x\\&\ge (a-\sigma _{5})\Vert \nabla u\Vert ^{2}_{2}t. \end{aligned}$$

For any $u\in S_{c}$ with $\Vert \nabla u\Vert ^{2}_{2}=1$ such that $t\star u\in {\mathcal {P}}_{c}$, by $Propositon \ 2.2$, we obtain $(\Psi _{u})^{\prime }(t)=0$, then

$$\begin{aligned} a-\sigma _{5}&=(a-\sigma _{5})\Vert \nabla u\Vert ^{2}_{2} \\&\le a\Vert \nabla u\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x t^{-2}\\&\le a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}t^{2}-\int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x t^{-2}\\&=3t^{-5}\int _{{\mathbb {R}}^{3}}{\tilde{G}}(t^{\frac{3}{2}}u)\textrm{d}x\\&=3t^{-5}\int _{{\mathbb {R}}^{3}}\left[ \frac{1}{2}g(t^{\frac{3}{2}}u)t^{\frac{3}{2}}u-G(t^{\frac{3}{2}}u)\right] \textrm{d}x\\&\le 3t^{-5}\left( \frac{1}{2}-\frac{1}{\beta }\right) \int _{{\mathbb {R}}^{3}}g(t^{\frac{3}{2}}u)t^{\frac{3}{2}}u\textrm{d}x\\&\le 3C_{1}t^{-5}\left( \frac{1}{2}-\frac{1}{\beta }\right) \left( \Vert t^{\frac{3}{2}}u\Vert ^{\alpha }_{\alpha }+\Vert t^{\frac{3}{2}}u\Vert ^{\beta }_{\beta }\right) \\&\le 3C_{1}C_{2}\left( \frac{1}{2}-\frac{1}{\beta }\right) \left[ t^{\frac{3}{2}\alpha -5}+t^{\frac{3}{2}\beta -5}\right] . \end{aligned}$$

Since $\beta>\alpha >\frac{14}{3}$ and (V5), we know that there exists a lower bound $\delta _{c}>0$. Hence, $\inf \limits _{u\in {\mathcal {P}}_{c}}\Vert \nabla u\Vert ^{2}_{2}\ge \delta ^{2}_{c}.$ $\square$

Next we apply the same decomposition of ${\mathcal {P}}_{c}$ to prove the following lemmas.

Lemma 4.3

Assume $(G1),(G3) \ and\ (V6)$ hold, then ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$ is closed in $H^{1}({\mathbb {R}}^{N})$.

Proof

For any $u\in {\mathcal {P}}$, we have

$$\begin{aligned} P[u]=a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{3}}W(x)u^{2}\textrm{d}x-3\int _{{\mathbb {R}}^{3}}{\tilde{G}}(u)\textrm{d}x=0. \end{aligned}$$

Notice that

$$\begin{aligned} (\Psi _{u})^{\prime }(t)=at\Vert \nabla u\Vert ^{2}_{2}+bt^{3}\Vert \nabla u\Vert ^{4}_{2}-t^{-1}\int _{{\mathbb {R}}^{N}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}x-3t^{-4}\int _{{\mathbb {R}}^{3}}{\tilde{G}}(t^{\frac{3}{2}}u)\textrm{d}x, \end{aligned}$$

this and (V6) yield immediately

$$\begin{aligned}&(\Psi _{u})^{\prime \prime }(1)\\&\quad =a\Vert \nabla u\Vert ^{2}_{2}+3b\Vert \nabla u\Vert ^{4}_{2}+\int _{{\mathbb {R}}^{3}}W(x)u^{2}\textrm{d}x+\int _{{\mathbb {R}}^{3}}\langle \nabla W(x),x\rangle u^{2}\textrm{d}x\\&\qquad +12\int _{{\mathbb {R}}^{3}}{\tilde{G}}(u)\textrm{d}x-\frac{9}{2}\int _{{\mathbb {R}}^{3}}{\tilde{G}}^{\prime }(u)u\textrm{d}x\\&\quad \le a\Vert \nabla u\Vert ^{2}_{2}+3b\Vert \nabla u\Vert ^{4}_{2}+\int _{{\mathbb {R}}^{3}}W(x)u^{2}\textrm{d}x+\int _{{\mathbb {R}}^{3}}\langle \nabla W(x),x\rangle u^{2}\textrm{d}x\\&\qquad +(12-\frac{9}{2}\alpha )\int _{{\mathbb {R}}^{3}}{\tilde{G}}(u)\textrm{d}x\\&\quad = a(5-\frac{3}{2}\alpha )\Vert \nabla u\Vert ^{2}_{2}+b(7-\frac{3}{2}\alpha )\Vert \nabla u\Vert ^{4}_{2}+(\frac{3}{2}\alpha -3)\int _{{\mathbb {R}}^{3}}W(x)u^{2}\textrm{d}x\\&\qquad +\int _{{\mathbb {R}}^{3}}\langle \nabla W(x),x\rangle u^{2}\textrm{d}x\\&\quad \le a(5-\frac{3}{2}\alpha )\Vert \nabla u\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{3}}Z(x)u^{2}\textrm{d}x\\&\quad \le -\left[ (\frac{3}{2}\alpha -5)a-\sigma _{6 }\right] \Vert \nabla u\Vert ^{2}_{2} <0, \end{aligned}$$

which implies ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$. Hence, one can see that ${\mathcal {P}}^{-}_{c}={\mathcal {P}}_{c}$ is closed in $H^{1}({\mathbb {R}}^{N})$. $\square$

Remark 4.4

By Lemma 2.3 and lemma 3.2, we know $J|_{{\mathcal {P}}^{-}_{c}}$ is a natural constraint of $J|_{S_{c}}.$

Remark 4.5

Suppose that $\left\{ u_{n}\right\} \subset {\mathcal {P}}^{-}_{c}$ such that $J[u_{n}]$ approaches a possible critical value. Since the constraint$(\Psi _{u_{n}})^{\prime \prime }(1)<0$ is open, there exist $\lambda _{n},\mu _{n}\in {\mathbb {R}}$ such that

$$\begin{aligned} J^{\prime }[u_{n}]+\lambda _{n}u_{n}+\mu _{n}P^{\prime }[u_{n}]\rightarrow 0. \end{aligned}$$

Then applying a similar argument as Lemma 2.3, we see that

$$\begin{aligned} \mu _{n}(\Psi _{u_{n}})^{\prime \prime }(1)\rightarrow 0. \end{aligned}$$

Notice that $(\Psi _{u_{n}})^{\prime \prime }(1)<0$, so

$$\begin{aligned} \mu _{n}\Vert \nabla u_{n}\Vert ^{2}_{2}\rightarrow 0. \end{aligned}$$

(4.7)

On the other hand, by Lemma 4.2, we obtain $\Vert \nabla u\Vert ^{2}_{2}\ge \delta _{c}^{2}>0$ for all $u\in {\mathcal {P}}_{c}$. Then, $\mu _{n}\rightarrow 0$. Thus if further $\{u_{n}\}$ is bounded in $H^{1}({\mathbb {R}}^{3})$, then we deduce that

$$\begin{aligned} J^{\prime }[u_{n}]+\lambda _{n}u_{n}\rightarrow 0 \ in \ H^{-1}({\mathbb {R}}^{3}). \end{aligned}$$

Corollary 4.6

Assume (G1)-(G3),(V4) and (V5) hold, then for any $u\in H^{1}({\mathbb {R}}^{3})\setminus \left\{ 0\right\}$, there exists an unique $t_{u}>0$ such that $t_{u}\star u\in {\mathcal {P}}$. Moreover,

$$\begin{aligned} J[t_{u}\star u]=\max \limits _{t>0}J[t\star u]. \end{aligned}$$

(4.8)

Proof

We first show the existence of $t_{u}$. For any $u\in H^{1}({\mathbb {R}}^{3})\setminus \left\{ 0\right\}$, there holds $\Vert \nabla u\Vert _{2}>0$. Let ${\tilde{m}}:=\Vert u\Vert ^{2}_{2}$, then by(V5), we obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}xt^{-1}&=\int _{{\mathbb {R}}^{3}}W(x)(t\star u)^{2}\textrm{d}xt^{-1}\\&\le \sigma _{5}\Vert \nabla (t\star u)\Vert ^{2}_{2}t^{-1}\\&=\sigma _{5}t\Vert \nabla u\Vert ^{2}_{2}. \end{aligned}$$

By (G1) and (G2), for $\forall t\ge 0$ and $s\in {\mathbb {R}}$, we have that

$$\begin{aligned} \left\{ \begin{aligned}&t^{\alpha } G(s)\le G(ts) \le t^{\beta } G(s), \ if \ t\le 1 \\&t^{\beta } G(s) \le G(ts)\le t^{\alpha } G(s), \ if \ t\ge 1. \end{aligned}\right. \end{aligned}$$

Then, it holds that

$$\begin{aligned} \frac{\alpha -2}{\beta -2}\min \left\{ t^{\alpha },t^{\beta }\right\} {\tilde{G}}(s)\le {\tilde{G}}(ts)\le \frac{\beta -2}{\alpha -2}\max \left\{ t^{\alpha },t^{\beta }\right\} {\tilde{G}}(s). \end{aligned}$$

Thus,

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}{\tilde{G}}(t^{\frac{3}{2}}u)\textrm{d}xt^{-4}&\le C_{3}(\Vert t^{\frac{3}{2}}u\Vert ^{\alpha }_{\alpha }+\Vert t^{\frac{3}{2}}u\Vert ^{\beta }_{\beta })t^{-4} \\&\le C_{3}(t^{\frac{3\alpha }{2}-4}\Vert u\Vert ^{\alpha }_{\alpha }+t^{\frac{3\beta }{2}-4}\Vert u\Vert ^{\beta }_{\beta }). \end{aligned}$$

and

$$\begin{aligned}&(\Psi _{u})^{\prime }(t)=a\Vert \nabla u\Vert ^{2}_{2}t+b\Vert \nabla u\Vert ^{4}_{2}t^{3}-\int _{{\mathbb {R}}^{3}}W\left( \frac{x}{t}\right) u^{2}\textrm{d}xt^{-1}-3\int _{{\mathbb {R}}^{3}}{\tilde{G}}(t^{\frac{3}{2}}u)\textrm{d}xt^{-4}\\&\quad \ge (a-\sigma _{5})\Vert \nabla u\Vert ^{2}_{2}t+b\Vert \nabla u\Vert ^{4}_{2}t^{3}- 3C_{3}\left( t^{\frac{3\alpha }{2}-4}\Vert u\Vert ^{\alpha }_{\alpha }+t^{\frac{3\beta }{2}-4}\Vert u\Vert ^{\beta }_{\beta }\right) . \end{aligned}$$

Since $\beta>\alpha >\frac{14}{3}$, one can see that,

$$\begin{aligned} (\Psi _{u})^{\prime }(t)>0, \ t\rightarrow 0. \end{aligned}$$

(4.9)

Therefore, there exists $t_{0}>0$ such that $\Psi _{u}(t)$ is increasing in $t \in (0,t_{0})$. On the other hand, by (V4), we have

$$\begin{aligned} \int _{{\mathbb {R}}^{3}}V(x)(t\star u)^{2}\le \sigma _{4}\Vert \nabla (t\star u)\Vert ^{2}_{2}=\sigma _{4}t^{2}\Vert \nabla u\Vert ^{2}_{2}. \end{aligned}$$

Thus,

$$\begin{aligned} \Psi _{u}(t)&=J[t\star u]\\&=\frac{a}{2}t^{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{3}}V(x)(t \star u)^{2}\textrm{d}x-t^{-3}\int _{{\mathbb {R}}^{3}}G(t^{\frac{3}{2}}u)\textrm{d}x\\&\le \frac{t^{2}}{2}(a+\sigma _{4})\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}-C_{4}t^{-3}(\Vert t^{\frac{3}{2}}u\Vert ^{\alpha }_{\alpha }+\Vert t^{\frac{3}{2}}u\Vert ^{\beta }_{\beta }) \\&=\frac{t^{2}}{2}(a+\sigma _{4})\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}t^{4}\Vert \nabla u\Vert ^{4}_{2}-C_{4}\left( t^{\frac{3\alpha }{2}-3}\Vert u\Vert ^{\alpha }_{\alpha }+t^{\frac{3\beta }{2}-3}\Vert u\Vert ^{\beta }_{\beta }\right) . \end{aligned}$$

Hence,

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }\Psi _{u}(t)=\lim \limits _{t\rightarrow +\infty }J[t\star u]=-\infty . \end{aligned}$$

(4.10)

By (4.10) (4.11), we deduce that there exists $t_{1}>t_{0}$ such that

$$\begin{aligned} J[t_{1}\star u]=\max \limits _{t>0}J[t\star u]. \end{aligned}$$

(4.11)

So $(\Psi _{u})^{\prime }(t_{1})=0$ and it follows proposition 2.2 that $t_{1}\star u\in {\mathcal {P}}$. Proof of the uniqueness of $t_{u}$ is similar to that of Corollary 3.6. We omit it. $\square$

Remark 4.7

Under the assumptions (G1)-(G3) and (V4)–(V6), for any $u\in {\mathcal {P}}_{c}$, we have that $\Psi _{u}(t)\rightarrow 0 \ as \ t\rightarrow 0^{+}$ and $\Psi _{u}(t)\rightarrow -\infty \ as \ t\rightarrow +\infty$. Corollary 4.6 implies that

$$\begin{aligned} J[u]=\max \limits _{t>0}J[t\star u]>0. \end{aligned}$$

For given $c>0$, define

$$\begin{aligned} M_{c}:=\inf \limits _{u\in {\mathcal {P}}_{c}}J[u]=\inf \limits _{u\in S_{c}}\max \limits _{t>0}J[t\star u]. \end{aligned}$$

(4.12)

Lemma 4.8

Under (G1)-(G3) and (V4)–(V6), $M_{c}>0$.

Proof

By (G2) and (V5), we have that

$$\begin{aligned} (a+\sigma _{5})\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}&\ge a\Vert \nabla u\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{N}}W(x)u^{2}\textrm{d}x+b\Vert \nabla u\Vert ^{4}_{2} \\&=3\int _{{\mathbb {R}}^{3}}{\tilde{G}}(u)\textrm{d}x\\&\ge \frac{3}{2}(\alpha -2)\int _{{\mathbb {R}}^{3}}G(u)\textrm{d}x. \end{aligned}$$

Then,

$$\begin{aligned} J[u]&=\frac{a}{2}\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}+\frac{1}{2}\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x-\int _{{\mathbb {R}}^{3}}G(u)\textrm{d}x \\&\ge \frac{1}{2}(a-\sigma _{4})\Vert \nabla u\Vert ^{2}_{2}+\frac{b}{4}\Vert \nabla u\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{3}}G(u)\textrm{d}x \\&\ge \left[ \frac{1}{2}(a-\sigma _{4})-\frac{2(a+\sigma _{5})}{3(\alpha -2)}\right] \Vert \nabla u\Vert ^{2}_{2}+\left[ \frac{b}{4}-\frac{2b}{3(\alpha -2)}\right] \Vert \nabla u\Vert ^{4}_{2}. \end{aligned}$$

Since $\alpha >\frac{14}{3}$, (V4) and (V5) imply $M_{c}>0$. $\square$

Corollary 4.9

Under (V1),(V4) and (V5), we have that

$$\begin{aligned} \lim \limits _{u\in {\mathcal {P}}_{c}, \Vert \nabla u\Vert ^{2}_{2}\rightarrow \infty }J[u]=+\infty . \end{aligned}$$

(4.13)

Proof

It follows by Lemma 4.8,

$$\begin{aligned} J[u]\ge \left[ \frac{1}{2}(a-\sigma _{4})-\frac{2(a+\sigma _{5})}{3(\alpha -2)}\right] \Vert \nabla u\Vert ^{2}_{2}+\left[ \frac{b}{4}-\frac{2b}{3(\alpha -2)}\right] \Vert \nabla u\Vert ^{4}_{2}, \end{aligned}$$

then the conclusion holds. $\square$

Proof of Theorem 1.2

Let $\{u_{n}\}\subset {\mathcal {P}}^{rad}_{c}$ be a minimizing sequence for J, i.e., $J[u_{n}]\rightarrow M_{c}$. Similar proof as that of Theorem 1.1, we get $M_{c}<m_{c}$. By Corollary 4.9, noting that $\{u_{n}\}$ is a bounded minimizing sequence in $H^{1}_{r}({\mathbb {R}}^{N})$, up to a subsequence, we may assume $u_{n}\rightharpoonup u$ in $H^{1}_{r}({\mathbb {R}}^{N})$, furthermore $u\ne 0$. Since $\{u_{n}\}$ is bounded in $H^{1}_{r}({\mathbb {R}}^{N})$, one can see that

$$\begin{aligned} \lambda _{n}=-\frac{\langle J^{\prime }[u_{n}],u_{n}\rangle }{c} \end{aligned}$$

is bounded. Meanwhile,

$$\begin{aligned} \lambda _{n}\Vert u_{n}\Vert ^{2}_{2}&=-\langle J^{\prime }[u_{n}],u_{n}\rangle \\&=\int _{{\mathbb {R}}^{3}}g(u_{n})u_{n}\textrm{d}x-a\Vert \nabla u_{n}\Vert ^{2}_{2}-b\Vert \nabla u_{n}\Vert ^{4}_{2}-\int _{{\mathbb {R}}^{3}}V(x)u^{2}_{n}\textrm{d}x \\&\ge \int _{{\mathbb {R}}^{3}}g(u_{n})u_{n}\textrm{d}x-\int _{{\mathbb {R}}^{3}}W(x)u^{2}_{n}\textrm{d}x-3\int _{{\mathbb {R}}^{3}}[\frac{1}{2}g(u_{n})u_{n}-G(u_{n})]\textrm{d}x \\&=3\int _{{\mathbb {R}}^{3}}G(u_{n})\textrm{d}x-\frac{1}{2}\int _{{\mathbb {R}}^{3}}g(u_{n})u_{n}\textrm{d}x-\int _{{\mathbb {R}}^{3}}W(x)u^{2}_{n}\textrm{d}x \\&\ge (3-\frac{\beta }{2})\int _{{\mathbb {R}}^{3}}G(u_{n})\textrm{d}x-\sigma _{5}\Vert \nabla u_{n}\Vert ^{2}_{2}. \end{aligned}$$

On the other hand,

$$\begin{aligned} (a-\sigma _{5})\Vert \nabla u_{n}\Vert ^{2}_{2}&\le (a-\sigma _{5})\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2} \\&\le a\Vert \nabla u_{n}\Vert ^{2}_{2}-\int _{{\mathbb {R}}^{N}}W(x)u^{2}_{n}\textrm{d}x+b\Vert \nabla u_{n}\Vert ^{4}_{2} \\&=3\int _{{\mathbb {R}}^{3}}\left[ \frac{1}{2}g(u_{n})u_{n}-G(u_{n})\right] \textrm{d}x \\&\le \frac{3(\beta -2)}{2}\int _{{\mathbb {R}}^{3}}G(u_{n})\textrm{d}x. \end{aligned}$$

Therefore,

$$\begin{aligned} \lambda _{n}\Vert u_{n}\Vert ^{2}_{2}\ge \left[ \left( 3-\frac{\beta }{2}\right) \frac{2(a-\sigma _{5})}{3(\beta -2)}-\sigma _{5}\right] \Vert \nabla u_{n}\Vert ^{2}_{2}. \end{aligned}$$

(V5) implies that

$$\begin{aligned} \left( 3-\frac{\beta }{2}\right) \frac{2(a-\sigma _{5})}{3(\beta -2)}-\sigma _{5}>0. \end{aligned}$$

Hence by Lemma 4.2, there exists $\delta >0$ such that $\lambda _{n}c>\delta$, $\forall n\in {\mathbb {N}}$. While $\{\lambda _{n}\}$ is bounded, up to a subsequence, we may suppose that there exists $\lambda >0$ such that $\lambda _{n}\rightarrow \lambda$.

Let assume $\Vert \nabla u_{n}\Vert ^{2}_{2}\rightarrow \theta$, then it is easy to see that u solves

$$\begin{aligned} -(a+b\theta )\Delta u+\lambda u+V(x)u=g(u) \ in \ {\mathbb {R}}^{3}. \end{aligned}$$

(4.14)

Since for all $u\in {\mathcal {P}}^{rad}_{c}$, there holds

$$\begin{aligned}&a\Vert \nabla u\Vert ^{2}_{2}+b\theta \Vert \nabla u\Vert ^{2}_{2}\\&=\frac{1}{2}\int _{{\mathbb {R}}^{3}}\langle \nabla V(x),x\rangle u^{2}\textrm{d}x-3\int _{{\mathbb {R}}^{3}}[\frac{1}{2}g(u)u-G(u)]\textrm{d}x \\&=\lim \limits _{n\rightarrow \infty }\left( \frac{1}{2}\int _{{\mathbb {R}}^{3}}\langle \nabla V(x),x \rangle u^{2}_{n}\textrm{d}x-3\int _{{\mathbb {R}}^{3}}[\frac{1}{2}g(u_{n})u_{n}-G(u_{n})]\textrm{d}x\right) \\&=\lim \limits _{n\rightarrow \infty }\left( a\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2}\right) =a\theta +b\theta ^{2}. \end{aligned}$$

So $(a+b\theta )(\Vert \nabla u\Vert ^{2}_{2}-\theta )=0$ and $a>0,b>0,\theta \ge 0$ imply $\Vert \nabla u\Vert ^{2}_{2}=\theta$. Hence, it follows by (4.14), u satisfies

$$\begin{aligned} -(a+b\int _{{\mathbb {R}}^{3}}|\nabla u|^{2}\textrm{d}x)\Delta u+\lambda u+V(x)u=g(u) \ in \ {\mathbb {R}}^{3}. \end{aligned}$$

Now we obtain that

$$\begin{aligned}&a\Vert \nabla u\Vert ^{2}_{2}+b\Vert \nabla u\Vert ^{4}_{2}+\lambda \Vert u\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{N}}V(x)u^{2}\textrm{d}x \\&=\int _{{\mathbb {R}}^{3}}g(u)u\textrm{d}x \\&=\int _{{\mathbb {R}}^{3}}g(u_{n})u_{n}\textrm{d}x+o(1) \\&=a\Vert \nabla u_{n}\Vert ^{2}_{2}+b\Vert \nabla u_{n}\Vert ^{4}_{2}+\lambda \Vert u_{n}\Vert ^{2}_{2}+\int _{{\mathbb {R}}^{N}}V(x)u^{2}_{n}\textrm{d}x+o(1), \end{aligned}$$

by which it follows that $\lambda (\Vert u_{n}\Vert ^{2}_{2}-\Vert u\Vert ^{2}_{2})=0$, and furthermore $\Vert u\Vert ^{2}_{2}=\Vert u_{n}\Vert ^{2}_{2}=c$. Hence, $u_{n}\rightarrow u \ in \ L^{2}_{r}({\mathbb {R}}^{N}),$ $u_{n}\rightarrow u \ in \ H^{1}_{r}({\mathbb {R}}^{N}),$ and $u\in {\mathcal {P}}^{rad}_{c}$. On the other hand, since $\int _{{\mathbb {R}}^{3}}g(u_{n})u_{n}\textrm{d}x\rightarrow \int _{{\mathbb {R}}^{3}}g(u)u\textrm{d}x$, $n\rightarrow \infty$, we have that

$$\begin{aligned} J[u]=\lim \limits _{n\rightarrow \infty }J[u_{n}]=M_{c}. \end{aligned}$$

Hence by lemma 2.3, there exists some $\lambda \in {\mathbb {R}}$ such that $(\lambda _{c},u_{c}):=(\lambda ,u)$ solves (1.1)–(1.2). That is, $(\lambda ,u)$ is a ground state normalized solution of Kirchhoff equation (1.1). $\square$