Normalized solutions for Kirchhoff-Carrier type equation

1.   Introduction and statement of results

In this paper, we study the following Kirchhoff-Carrier type equation

where $a, \ b > 0$ are constants, $\lambda\in \mathbb{R}, \ 2 < p < \tau < 6$. $M$ satisfies the following condition

(H) $M: \mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ is continuous and

for some $C_{0}, \ d_{1}, \ d_{2} > 0$. Moreover, for each $\sigma\in [0, S_{\tau}], \ t\geq 0,$

where $S_{\tau}$ is the best constant of $H^{1}(\mathbb{R}^{3})$ embedding into $L^{\tau}(\mathbb{R}^{3})$.

It is well known that problem (1.1) comes from two classes of typical nonlocal problem which are Kirchhoff type and Carrier type problems. In 1883, Kirchhoff ^[10] firstly proposed the Kirchhoff type problem

which extends the claasical D'Alembert wave equation by considering the effects of changes in the length of the strings during the vibrations. In ^[12], Lions proposed a functional analysis method to solve the following Kirchhoff problem

Since then, problem (1.4) has attracted the attention of researchers, see ^{[4,5,14,15,19,20]} and the references therein.

Carrier in ^[7] proposed the following problem

The vibration of elastic string is described by (1.5) when the tension change is not very small. We note that problem (1.5) is similar to the Kirchhoff equation which has many research results. However, there is very little work on the problem (1.5) or its generalization. The appearance of nonlocal terms leads to some difficulties. It lacks a variational structure so we cannot study the problem (1.5) with variational method. Some authors are concerned about the existence of positive solutions or some generalized cases of problem (1.5) by the topological theory and pseudo montone operator theory, see ^[1,6]. It is worth mentioning that in ^[9], Jin and Yan studied the following Carrier type problem

where $a, b > 0$ and $\Omega\subset \mathbb{R}^{N}$ is a bounded open set with smooth boundary and $\gamma\geq 1$. They obtained the existence of sign-changing solutions of problem (1.6) by using the fixed point index method. Xu and Qin ^[21] considered the Kirchhoff-Carrier type equation

where $a, b > 0$ and $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary and $1 < \gamma < 2^*, \ \alpha: \mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow\; \mathbb{R}^{+}$. By applying the mountain pass theorem and the Ekeland theorem, they obtained some existence results for problem (1.7). Let us emphasize that all the previous existence results for Kirchhoff-Carrier type equation are obtained in a bounded domain. Obviously, the approaches adopted in ^[9,21] do not work when the Kirchhoff-Carrier type problem defined on the whole space $\mathbb{R}^{3}$.

When $b = 0$, adding a repulsive nonlocal Coulombic potential, problem (1.1) reduces to the Schrödinger-Poisson-Slater equation

Luo ^[13] proved a multiplicity result of solutions for problem (1.8) if $p\in (\frac{10}{3}, 6)$. They were interested in normalized solutions, that is, solutions to (1.8) satisfying

The normalized solutions associated to Schrödinger or Kirchhoff problems have been extensively studied in recent years, see ^[11,17,22]. Because people are particularly interested in normalized solutions, we can search for normalized solutions of (1.1). Precisely, for given $c > 0$ we look for

For this reason, we construct the variational structure of problem (1.1). Define $J_{b}: H_{r}^{1}(\mathbb{R}^{3})\rightarrow \mathbb{R}$ is a functional with its Fréchet derivative operator given by

For any $u\in H_{r}^{1}(\mathbb{R}^{3})\setminus \{0\}$ set $\beta_{u} = \frac{u}{|\nabla u|_{2}}$ and $v(t) = t\beta_{u}$ for all $t\geq 0$. It follows from (1.9) that

Integrating over $[0, |\nabla u|_{2}]$, using Fubini's theorem and $J_{b}(0) = 0$ we obtain

for $u\in H_{r}^{1}(\mathbb{R}^{3})$. Since $M$ is continuous, $J_{b}\in \mathcal{C}^{1}(H_{r}^{1}(\mathbb{R}^{3}), \mathbb{R})$ and the couple of weak solution as above can be viewed as a critical point of $J_{b}$ restricted on the constraint

As far as we know, there is no result on the existence and asymptotic behaviour of normalized solutions of problem (1.1). In this paper, we focus on the existence of high energy normalized solutions to (1.1). Let us state our main result.

Theorem 1.1. Let $\max\left\{\frac{2}{3}(d_{1}+2)+2, \ \frac{2}{3}(d_{2}+2)+2\right\} < p < 6$. Then, for any fixed $c > 0$ (1.1) has a sequence of couples of weak solutions $\{(v_{n}, \lambda_{n})\}\subset H^{1}_{r}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ with $|v_{n}|_{2}^{2} = c$ for each $n\in \mathbb{N}^{+},$

Theorem 1.1 shows $(v_{n}, \lambda_{n})$ depends on the parameter $b$. Therefore, we next focus on whether some converence phenomena appear when $b\rightarrow 0$.

Theorem 1.2. Let $\max\left\{\frac{2}{3}(d_{1}+2)+2, \ \frac{2}{3}(d_{2}+2)+2\right\} < p < 6$. $\{(v_{n}^{b}, \lambda_{n}^{b})\}\subset X_{r}(c)\times \mathbb{R}^{-}$ are obtained in Theorem 1.1. Then, for any sequence $\{b_{m}\}\rightarrow 0^{+}(m\rightarrow +\infty)$ there exists a subsequence still denoted by $\{b_{m}\}$ such that for any $n\in \mathbb{N}^{+}$ $v_{n}^{b_{m}}\rightarrow v_{n}^{0}$ in $H_{r}^{1}(\mathbb{R}^{3})$ and $\lambda_{n}^{b_{m}}\rightarrow \lambda_{n}^{0}$ in $\mathbb{R}$ as $m\rightarrow +\infty$ where $\{(v_{n}^{0}, \lambda_{n}^{0})\}\subset X_{r}(c)\times \mathbb{R}^{-}$ is a sequence of couples of weak solutions to the following equation

Remark 1.1. A typical example of $M$ satisfying condition (H) is $M(s, t) = a+2bt^{2}$ with $a, b > 0$.

The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries results. In Section 3, we prove Theorems 1.1 and 1.2.

2.   Preliminaries

Throughout the paper, let $\|u\| = \left(\int_{ \mathbb{R}^{3}}(|u|^{2}+|\nabla u|^{2})dx\right)^{\frac{1}{2}}$ denote the usual norm of $H^{1}(\mathbb{R}^{3})$, $|\cdot|_{\tau}$ denote the norm of $L^{\tau}(\mathbb{R}^{3})$ for $2 < \tau < 6$. $H^{-1}(\mathbb{R}^{3})$ is the dual space of $H^{1}(\mathbb{R}^{3})$ and $H_{r}^{1}(\mathbb{R}^{3})$ is the subspace of radially symmetric functions in $H^{1}(\mathbb{R}^{3})$.

Proposition 2.1. [3,Lemma 2.1] Assume that $p\in (2, 6).$ Then, there holds

where $\{V_{n}\}\subset H_{r}^{1}(\mathbb{R}^{3})$ be a strictly increasing sequence of finite-dimensional linear subspace in $H_{r}^{1}(\mathbb{R}^{3})$ such that $\cup_{n}V_{n}$ is dense in $H_{r}^{1}(\mathbb{R}^{3})$ and $V_{n}^{\perp}$ denotes the orthogonal complementary space of $V_{n}$ in $H_{r}^{1}(\mathbb{R}^{3})$.

This lemma is useful to show the compactness of the Palais-Smale sequence.

Lemma 2.2. [2,Lemma 3] Let $F$ be a $\mathcal{C}^{1}$ functional on $H^{1}(\mathbb{R}^{3})$, if $\{x_{k}\}\subset X(c)$ is bounded in $H^{1}(\mathbb{R}^{3})$. Then,

where $X(c) = \left\{u\in H^{1}(\mathbb{R}^{3}):\int_{ \mathbb{R}^{3}}|u_{c}|^{2}dx = c\right\}, \quad c > 0.$

Set

Now, we introduce a scaling. Define $u_{t}(x) = t^{\frac{3}{2}}u(tx)$, for $t > 0, \ u\in X_{r}(c)$. Then,

Lemma 2.3. Let $p > \max\left\{\frac{2}{3}(d_{1}+2)+2, \frac{2}{3}(d_{2}+2)+2\right\}, c > 0$ and $u\in X_{r}(c)$. Then

Proof. It follows from $D(u_{t}) = D(u)$ that $u_{t}\in X_{r}(c)$ for any $u\in X_{r}(c)$. By (1.2) and (1.10), we obtain

which implies that $J_{b}(u_{t})\rightarrow -\infty,$ as $t\rightarrow +\infty$ since $\tau\in (2, 6)$ and $p > \max\left\{\frac{2}{3}(d_{1}+2)+2, \frac{2}{3}(d_{2}+2)+2\right\}$. □

For $c > 0$ fixed, $n\in \mathbb{N}^{+}$ and $n\geq 2$ let

and

where $L = \max_{x > 0}\frac{(x^{2}+c)^{p/2}}{x^{p}+c^{p/2}}, \ \mu_{n}$ is given in Proposition 2.1.

Lemma 2.4. For any $p\in (2, 6)$, then $\sigma_{n}\rightarrow +\infty$ as $n\rightarrow +\infty$. Particularly, we can suppose that $\sigma_{n}\geq 1$ for every $n\in \mathbb{N}^{+}$ without any restriction.

Proof. For any $u\in E_{n}$, we deduce that

Combining with Proposition 2.1, we obtain that $\sigma_{n}\rightarrow +\infty$ as $n\rightarrow +\infty$ since $p > 2$. □

Define the continuous map

It follows from Lemma 2.3 that $\eta(t, v)\in X_{r}(c)$ for any $v\in X_{r}(c), \ t\in \mathbb{R}$ and

Recalling that $V_{n}$ is finite dimensional, we obtain that for any $n\in \mathbb{N}^{+}$, there exists $t_{n} > 0$ such that

satisfies

Now, define

Obviously, $\bar{\gamma}_{n}\in \Gamma_{n}$. According to [3,Lemma 2.3], we obtain the next key intersection result.

Lemma 2.5. For each $n\in \mathbb{N}^{+}$,

Proof. The proof of the lemma can be easily obtained by [3,Lemma 2.3]. □

Next, we will show that $\{\kappa_{n}^{b}(c)\}$ is indeed a sequence of critical values of $J_{b}$ restricted on $X_{r}(c)$. First, we shall prove that there exists a bounded (PS) sequence at each level $\kappa_{n}^{b}(c)$. Now, we fix an $n\in \mathbb{N}^{+}$. We define the following auxiliary functional:

where $\eta(t, u)$ is defined in (2.1). Set

Obviously, $\tilde{\gamma}: = (0, \gamma)\in \tilde{\Gamma}_{n}$ for any $\gamma\in \Gamma_{n}$. Define

Applying the fact that the maps

and

satisfy

we get $\tilde{\kappa}_{n}^{b}(c) = \kappa_{n}^{b}(c)$. Let us denote by $E$ the space $\mathbb{R} \times H_{r}^{1}(\mathbb{R}^{3})$ endowed with the norm $\|\cdot\|_{E}^{2} = |\cdot|^{2}_{ \mathbb{R}}+\|\cdot\|^{2}$ and by $E^{*}$ its dual space. Similar to [8,Lemma 2.3], we get the following result.

Lemma 2.6. For any $\epsilon > 0$, suppose that $\tilde{\gamma}_{0}\in \tilde{\Gamma}_{n}$ satisfies

Then, there exists a pair of $(t_{0}, u_{0})\in \mathbb{R}\times X_{r}(c)$ such that

(i) $\tilde{J}_{b}(t_{0}, u_{0})\in [\tilde{\kappa}_{n}^{b}(c)-\epsilon, \tilde{\kappa}_{n}^{b}(c)+\epsilon]$;

(ii) $\min_{0\leq s\leq 1, u\in X_{r}(c)\cap V_{n}}\|(t_{0}, u_{0})-\tilde{\gamma}_{0}(s, u)\|_{E}\leq \sqrt{\epsilon};$

(iii) $\left\|\left(\tilde{J}_{b}|_{X_{r}(c)\times \mathbb{R}}\right)'(t_{0}, u_{0})\right\|_{E^{*}}\leq 2\sqrt{\epsilon}$, i.e.,

holds, for all $z\in \tilde{T}_{(t_{0}, u_{0})}: = \{(z_{1}, z_{2})\in E, \langle u_{0}, z_{2}\rangle_{L^{2}} = 0\}$.

Lemma 2.7. For any $c > 0$ fixed and $n\in \mathbb{N}^{+}$, there exists a sequence $\{v_{k}^{n}\subset X_{r}(c)\}$ satisfying

where

Particularly, $\{v_{k}^{n}\}$ is bounded in $X_{r}(c)$.

Proof. By the definition of $\kappa_{n}^{b}(c)$ we have that for every $k\in \mathbb{N}^{+}$, there exists a $\gamma\in \Gamma_{n}$ such that

It follows from $\tilde{\kappa}_{n}^{b}(c) = \kappa_{n}^{b}(c)$ and $\tilde{\gamma}_{k}: = (0, \gamma_{k})\in \tilde{\Gamma}_{n}$ that

Using Lemma 2.6, we infer that there exists a sequence $\{(t_{k}^{n}, u_{k}^{n})\}\subset \mathbb{R}\times X_{r}(c)$ such that

(i) $\tilde{J_{b}}(t_{k}^{n}, u_{k}^{n})\in [\kappa_{n}^{b}(c)-\frac{1}{k}, \kappa_{n}^{b}(c)+\frac{1}{k}]$;

(ii) $\min_{0\leq s\leq 1, u\in X_{r}(c)\cap V_{n}}\|(t_{k}^{n}, u_{k}^{n})-(0, \gamma_{k})\|_{E}\leq \sqrt{\frac{1}{k}}$;

(iii) $\left\|\left(\tilde{J}_{b}|_{X_{r}(c)\times \mathbb{R}}\right)'(t_{k}^{n}, u_{k}^{n})\right\|_{E^{*}}\leq \frac{4}{\sqrt{k}}$, i.e.,

holds, for all $z\in \tilde{T}_{(t_{k}^{n}, u_{k}^{n})}: = \{(z_{1}, z_{2})\in E, \langle u_{k}^{n}, z_{2}\rangle_{L^{2}} = 0\}$.

For every $k\in \mathbb{N}^{+}$, set $v_{k}^{n} = \eta(t_{k}^{n}, u_{k}^{n})$. We shall show that $\{v_{k}^{n}\}\subset X_{r}(c)$ satisfies (2.2). It follows from (i) that $J_{b}(v_{k}^{n})\rightarrow \kappa_{n}^{b}(c)$ as $k\rightarrow \infty$, since $J_{b}(v_{k}^{n}) = J_{b}(\eta(t_{k}^{n}, u_{k}^{n})) = \tilde{J_{b}}(t_{k}^{n}, u_{k}^{n})$. Noting that

we obtain

Hence, by (iii), we see that $G_{b}(v_{k}^{n})\rightarrow 0$ as $k\rightarrow \infty$ for $(1, 0)\in \tilde{T}_{(t_{k}^{n}, u_{k}^{n})}$.

Finally, we shall show that

We claim that for $k\in \mathbb{N}$ large enough

holds for all $w\in T_{v_{k}^{n}}$, where $T_{v_{k}^{n}}: = \{w\in H_{r}^{1}(\mathbb{R}), \langle v_{k}^{n}, w\rangle_{L^{2}} = 0\}$. Indeed, for $w\in T_{v_{k}^{n}}$, taking $\tilde{w} = \eta(-t_{k}^{n}, w)$ we have

From $\int_{ \mathbb{R}^{3}}u_{k}^{n}\tilde{w}dx = \int_{ \mathbb{R}^{3}}v_{k}^{n}wdx$, we see that $(0, \tilde{w})\in \tilde{T}_{(t_{k}^{n}, u_{k}^{n})}$ is equivalent to show $w\in T_{v_{k}^{n}}$. It follows from (ii) that

Hence, we conclude that

Therefore,

which yields

as $k\rightarrow \infty$. Since (1.3) and $p\in (\frac{10}{3}, 6)$, we obtain

which completes the proof. □

Lemma 2.8. Let $p\in (2, 6), \ \lambda\in \mathbb{R}$. If $u\in H^{1}(\mathbb{R}^{3})$ is a weak solution of

then $G_{b}(u) = 0$. Moreover if $\lambda\geq 0$, we obtain $u = 0$.

Proof. If $u\in H^{1}(\mathbb{R}^{3})$ is a weak solution of (1.1) then it satisfies the Pohožaev identity

and

Thus, we obtain

Combining (2.4) and (2.5), we can see that

Since $p\in (2, 6)$, if $\lambda > 0$, we obtain $u\equiv 0$ immediately. If $\lambda = 0$, we get $A(u) = 0$. Together with $G_{b}(u) = 0$ then $u\equiv 0$. □

Lemma 2.9. Let $v_{k}^{n}\subset X_{r}(c)$ be the Palais-Smale sequence obtained in Lemma 2.7. Then, there exist $v_{n}\in H_{r}^{1}(\mathbb{R}^{3})$ and $\{\lambda_{k}^{n}\}\subset \mathbb{R}$ such that up to a subsequence as $k\rightarrow +\infty$

(i) $v_{k}^{n}\rightharpoonup v_{n}\neq 0$ in $H_{r}^{1}(\mathbb{R}^{3})$;

(ii) $\lambda_{k}^{n}\rightarrow \lambda_{n} < 0$ in $\mathbb{R}$;

(iii) $-\left(a+bM(|\nabla v_{k}^{n}|_{2}, |\nabla v_{k}^{n}|_{2}|\beta_{v_{k}^{n}}|_{\tau})\right)\Delta v_{k}^{n}-\lambda_{k}^{n} v_{k}^{n}-|v_{k}^{n}|^{p-2}v_{k}^{n}\rightarrow 0$ in $H_{r}^{1}(\mathbb{R}^{3})$;

(iv) $-\left(a+bM(|\nabla v_{n}|_{2}, |\nabla v_{n}|_{2}|\beta_{v_{n}}|_{\tau})\right)\Delta v_{n}-\lambda_{n} v_{n}-|v_{n}|^{p-2}v_{n} = 0$ in $H_{r}^{-1}(\mathbb{R}^{3})$.

Moreover, if $\lambda_{n} < 0$, then we obtain

Proof. Since $\{v_{k}^{n}\}\subset X_{r}(c)$ is bounded, we may assume that there exists $v_{n}\in H_{r}^{1}(\mathbb{R}^{3})$ such that

We claim that $v_{n}\neq 0$. In fact, we assume by contradiction that $v_{n} = 0$, then $C(v_{n}^{k}) = o(1)$. By $G_{b}(v_{k}^{n}) = o(1)$, we obtain $A(v_{k}^{n}) = o(1)$. Consequently, $J_{b}(v_{k}^{n}) = o(1)$ which contradicts Lemma 2.4. Hence, (i) is proved. It follows from Lemma 2.2 that

Note that for any $w\in H_{r}^{1}(\mathbb{R}^{3})$,

where

Therefore, (iii) is proved. By (2.6) and the fact $\{v_{k}^{n}\}\subset X_{r}(c)$ is bounded up to a subsequence, there exists $\lambda_{n}\in \mathbb{R}$ such that $\lambda_{k}^{n}\rightarrow \lambda_{n}$ as $k\rightarrow +\infty$. Moreover, from $2 < p < 6$ and (2.5), we deduce that

which yields $\lambda_n\leq 0$. Consequently, it follows from Lemma 2.8 that $\lambda_{n} < 0$, that is, (ii) is true.

Assume that $\lim_{k\rightarrow \infty}\int_{ \mathbb{R}^{3}}|\nabla v_{k}^{n}|^{2}dx = l^{2}\geq 0.$ By (ii) and (iii), we obtain

From (2.8), we get

From (ii) and (iii), we obtain

By (2.9) and (2.10), we see that

Recalling that $v_{k}^{n}\rightarrow v_{n}$ in $L^{p}(\mathbb{R}^{3})$, we deduce that

which yields that $\left(a+bM(l, |v_{n}|_{\tau})\right)\int_{ \mathbb{R}^{3}}|\nabla v_{k}^{n}-\nabla v_{n}|^{2}dx = o(1)$ for $a, \ b > 0, l\geq 0, \lambda_{n}\leq 0$. Hence,

Together with (ii) and (iii), (iv) is easily obtained. Then, we obtain that $\int_{ \mathbb{R}^{3}}|v_{k}^{n}-v_{n}|^{2}dx = o(1)$. Thus, we obtain

□

3.   Proofs of the main results

Proof of Theorem 1.1. From Lemma 2.9, we obtain that for any fixed $c > 0$, $\{(v_{n}, \lambda_{n})\}\subset H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ with $|v_{n}|_{2}^{2} = c$ is a sequence of couples of weak solutions to (1.1) for each $n\in \mathbb{N}^{+}$. Since $G_{b}(v_{n}) = 0$, (1.2) and (2.3) we obtain

which implies that $\{v_{n}\}$ is unbounded due to Lemmas 2.4 and 2.5. □

Proof of Theorem 1.2. For every $b > 0$, by Theorem 1.1 there exists a sequence of couples of weak solutions $\{(v_{n}^{b}, \lambda_{n}^{b})\}\subset H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ to (1.1). We show that for any sequence $b_{m}\rightarrow 0^{+}$ as $m\rightarrow +\infty$, $\{v_{n}^{b_{m}}\}_{m\in \mathbb{N}^{+}}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$. It follows from $V_{n}$ is finite-dimensional that for each $n\in \mathbb{N}^{+}$,

Since $\{(v_{n}^{b_{m}}, \lambda_{n}^{b_{m}})\}\subset H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ is a sequence of couples of weak solutions to (1.1) with $b = b_{m}$ and

then by Lemma 2.8, we conclude that $G_{b_{m}}(v_{n}^{b_{m}}) = 0$. Together with (1.3) and (2.3), we deduce that

which implies that $\{A(v_{n}^{b_{m}})\}_{m\in \mathbb{N}^{+}}$ is bounded in $\mathbb{R}$, $\{v_{n}^{b_{m}}\}_{m\in \mathbb{N}^{+}}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$ and $\{\lambda_{n}^{b_{m}}\}_{m\in \mathbb{N}^{+}}$ is bounded in $\mathbb{R}$. Therefore, there exist a subsequence of $\{b_{m}\}$ (still denoted by $\{b_{m}\}$) and $\lambda_{n}^{0}\leq 0$ such that $\lambda_{n}^{b_{m}}\rightarrow \lambda_{n}^{0}$ as $m\rightarrow +\infty$. Meanwhile,

Thus, for every $n\in \mathbb{N}^{+}$ $(v_{n}^{0}, \lambda_{n}^{0})$ is a couple of weak solution to (1.1) with $b = 0$, that is, for any $w\in H_{r}^{1}(\mathbb{R}^{3})$,

Taking $w = v_{n}^{b_{m}}-v_{n}^{0}$ in (3), we obtain

By $\{(v_{n}^{b_{m}}, \lambda_{n}^{b_{m}})\}\subset H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ is a sequence of couples of weak solutions to (1.1) with $b = b_{m}$, $\lambda_{n}^{b_{m}}\rightarrow \lambda_{n}^{0}$ and $\{v_{n}^{b_{m}}\}$ is bounded in $H_{r}^{1}(\mathbb{R}^{3})$, we obtain

Combining (3.1), (3.3) and (3.4) we can see that

According to $\lambda_{n}^{0}\leq 0$, $A(v_{n}^{b_{m}}-v_{n}^{0})\rightarrow 0$ as $m\rightarrow +\infty$. If $\lambda_{n}^{0} = 0$, from (3.2) it follows that $v_{n}^{0}\in H_{r}^{1}(\mathbb{R}^{3})$ is a weak solution to $-a\Delta v = |v|^{p-2}v$. This yields $v_{n}^{0} = 0$. On the other hand, by Lemma 2.5 we obtain that

which yields a contradiction. Hence, $\lambda_{n}^{0} < 0$ and $D(v_{n}^{b_{m}}-v_{n}^{0})\rightarrow 0$, as $m\rightarrow +\infty,$ due to (3.5). Thus, $\{(v_{n}^{0}, \lambda_{n}^{0})\}\subset H_{r}^{1}(\mathbb{R}^{3})\times \mathbb{R}^{-}$ is a sequence of couples of weak solutions to (1.11). □

4.   Conclusions

The main purpose of this paper is to study the existence of high energy normalized solutions for the Kirchhoff-Carrier type equation. To prove Theorem 1.1, we first show that

Combining with Proposition 2.1, we obtain that $\sigma_{n}\rightarrow +\infty$ as $n\rightarrow +\infty$. Similar to [8,Lemma 2.3], we get Lemma 2.6. We use Lemma 2.6 to obtain a Palais-Smale sequence $\{v_{k}^{n}\subset X_{r}(c)\}$ of functional $J_{b}$ satisfying $G_{b}(v_{k}^{n})\rightarrow 0$ where $G_{b}(u)$ is given in Lemma 2.7. Finally, we conclude that

by using Lemma 2.2. Meanwhile, we find some converence phenomena appear when $b\rightarrow 0$.

In the proof, $H_{r}^{1}(\mathbb{R}^{3})$ is the subspace of radially symmetric functions in $H^{1}(\mathbb{R}^{3})$ is very crucial, we do not know whether the solution can still exist in the more general space $H^{1}(\mathbb{R}^{3})$. This is a question that we need to further consider.

Use of AI tools declaration

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

Acknowledgments

J. Yang is supported by Natural Science Foundation of Hunan Province of China (No. 2023JJ30482, 2022JJ30463), Research Foundation of Education Bureau of Hunan Province (No. 22A0540) and Huaihua University Double First-Class Initiative Applied Characteristic Discipline of Control Science and Engineering. H. B. Chen is supported by the National Natural Science Foundation of China (No. 12071486).

Conflict of interest

The authors declare no conflict of interest.