Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations

Abstract

In this paper, we consider the $L^2$-norm prescribed ground states of the Kirchhoff equations involving mass critical exponent with a Lagrange multiplier $\mu \in {\mathbb {R}}$,

$$\begin{aligned} \left\{ \begin{aligned}&-\left( a+b\int _{{\mathbb {R}}^N} |\nabla u|^2dx\right) \Delta u+V(x)u=\mu u+u^{\frac{8}{N}+1}\quad \text {in }{\mathbb {R}}^N,\\&\int _{{\mathbb {R}}^N} |u|^2dx=c^2, \;\;c>0, \end{aligned} \right. \end{aligned}$$

(0.1)

where $a\ge 0$, $b>0$, $N=1,2,3$, and the function $V(x)\in L_{loc}^\infty ({\mathbb {R}}^N)$ is a trapping potential satisfying $\min _{x\in {\mathbb {R}}^N} V(x)=0$, $V(x)\rightarrow +\infty$ as $|x|\rightarrow +\infty$. It has been shown by researchers that there exists a couple of ground state solution $(u_a, \mu _a)$ to (0.1) if $c=c_*:=(\frac{b\Vert Q\Vert _2^{8/N}}{2})^{\frac{N}{8-2N}}$ for small $a>0$, where $Q>0$ is the unique radially symmetric positive solution of equation $2\Delta Q+\frac{N-4}{N}Q+Q^{\frac{8}{N}+1}=0$ in ${\mathbb {R}}^N$. We devote to the refined limiting profiles of $u_a$ as $a\rightarrow 0$ by using energy estimates and blow-up analysis. In order to get the concentration behavior of $u_a$, we first study the existence and non-existence of solutions to a degenerate Kirchhoff equation i.e. the case $a=0$ in (0.1). At last, we investigate the local uniqueness of ground states $u_a$ included by concentration.

Appendix A

We give the rigorous definition of the ground states of (1.1) with fixed $a>0$ and $\mu <0$. First of all, the associated energy functional $J_{a,\mu }(u):{\mathcal {H}}\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} J_{a,\mu }(u)= & {} \frac{1}{2}\int _{{\mathbb {R}}^N}a|\nabla u|^2+\left( V(x)-\mu \right) u^2dx+\frac{b}{4}\left( \int _{{\mathbb {R}}^N}|\nabla u|^2dx\right) ^2\\&-\frac{1}{2q+2}\int _{{\mathbb {R}}^N}|u|^{2q+2}dx. \end{aligned}$$

If $u\in {\mathcal {H}}\setminus \{0\}$ satisfies $\langle J_{a,\mu }'(u), \varphi \rangle =0$ for any $\varphi \in {\mathcal {H}}$, then u is a nontrivial weak solution to (1.1). Define

$$\begin{aligned} {\mathcal {S}}_{a,\mu }=\{u(x): \;u\text { is a nontrivial solution to } (1.1)\text { with fixed }a>0, \mu <0\}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {G}}_{a,\mu }=\{u(x)\in {\mathcal {S}}_{a,\mu }:\; J_{a,\mu }(u)\le J_{a,\mu }(v)\text { for any }v\in {\mathcal {S}}_{a,\mu }\}. \end{aligned}$$

(A.1)

Therefore, we say $u\in {\mathcal {H}}$ is a ground state solution to (1.1) if $u\in {\mathcal {G}}_{a,\mu }$. According to the constraint minimization problem (1.2) in the case $c=c_*$, we denote by ${\mathcal {M}}_{a}$ the collection of all minimizers,

$$\begin{aligned} {\mathcal {M}}_{a}=\{u\in {\mathcal {H}}:\;u\text { is a minimizer of } e(a,c_*)\}. \end{aligned}$$

(A.2)

Under the assumptions $(V_2)$ and $(V_3)$, Theorem 1.3 shows that there exists $a_*>0$, such that if $0<a<a_*$ the set $\mathcal M_{a}$ contains only one component $u_a$ satisfying with the Lagrange multiplier $\mu _a<0$,

$$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^N}|\nabla u_a|^2dx\right) \Delta u_a+V(x)u_a=\mu _a u_a+u_a^{2q+1}\quad \hbox { in}\ {\mathbb {R}}^N. \end{aligned}$$

(A.3)

Now, we adapt a similar argument in [19] to show the relationship ${\mathcal {M}}_a={\mathcal {G}}_{a,\mu _a}$ for $0<a<a_*$. This can be viewed as a interesting application of the local uniqueness of $u_a$. Let u be a ground state solution of (1.1) with $\mu =\mu _a$, we know that

$$\begin{aligned} \begin{aligned} E_a(u)=&2J_{a,\mu _a}(u)+ \mu _a\int _{{\mathbb {R}}^N}|u|^2dx\\ 4J_{a,\mu _a}(u)=&\int _{{\mathbb {R}}^N}a|\nabla u|^2+\left( V(x)-\mu _a\right) u^2dx+\frac{q-1}{q+1}\int _{{\mathbb {R}}^N}|u|^{2q+2}dx. \end{aligned} \end{aligned}$$

(A.4)

Set ${\bar{u}}=\rho u$ where $\rho =\frac{c_*}{\Vert u\Vert _2}$, then $\int _{{\mathbb {R}}^N}|{\bar{u}}|^2dx=c_*^2$. Since $u_a$ is a minimizer of $e(a,c_*)$ and $u\in {\mathcal {G}}_{a,\mu _a}$, we obtain that

$$\begin{aligned} E_{a}(u_a)\le E_a({\bar{u}})\quad \text {and}\quad J_{a,\mu _a}(u)\le J_{a,\mu _a}(u_a). \end{aligned}$$

From (A.4) one has

$$\begin{aligned} J_{a,\mu _a}(u)\le J_{a,\mu _a}(u_a)\le J_{a,\mu _a}(\bar{u})=J_{a,\mu _a}(\rho u). \end{aligned}$$

(A.5)

On the other hand, we define

$$\begin{aligned} \begin{aligned} f(\rho ):=J_{a,\mu _a}(\rho u)=&\frac{\rho ^2}{2}\int _{{\mathbb {R}}^N}a|\nabla u|^2+\left( V(x)-\mu _a\right) u^2dx\\&+\frac{b\rho ^4}{4}\left( \int _{{\mathbb {R}}^N}|\nabla u|^2dx\right) ^2-\frac{\rho ^{2q+2}}{2q+2}\int _{{\mathbb {R}}^N}|u|^{2q+2}dx.\\ \end{aligned} \end{aligned}$$

It’s not hard to see $f(\rho )\in C(0,+\infty )$ has only one maximum point at $\rho =1$. Hence we deduce from (A.5) that $\Vert u\Vert _2=c^*$ and $u\in {\mathcal {M}}_a$.