Normalized solutions to the Kirchhoff equation with $L^2$-subcritical or critical nonlinearities in ${\varvec{\mathbb {R}}}^{2}$

Abstract

In this paper, we study the existence of normalized solutions to the Kirchhoff equation with $L^2$-subcritical or critical nonlinearities

$$\begin{aligned} \begin{array}{ll} - \left( {a + b\int _{{\mathbb {R}}^{2}} {|\nabla u{|^2}dx} } \right) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~\textrm{in}~{\mathbb {R}}^{2}, \end{array} \end{aligned}$$

where $a,b,\mu >0$, $2< q < p \le 6$. By minimizing methods and the concentration compactness principle, we prove the existence and nonexistence of normalized solutions when (p, q) belongs to a certain domain in ${{\mathbb {R}}^{2}}$, and discuss how $\mu$ affects the existence of normalized solutions. Our main results may be illustrated by the red areas and green areas shown in Figure 1. Our results partially extend the results of Soave (J. Differ. Equ. 2020).

1 Introduction

In this paper, we study the existence of normalized solutions to the Kirchhoff equation with $L^2$-subcritical or critical nonlinearities

$$\begin{aligned} \begin{array}{ll} - \left( {a + b\int _{{\mathbb {R}}^{2}} {|\nabla u{|^2}dx} } \right) \Delta u = \lambda u + |u{|^{p - 2}}u + \mu |u{|^{q - 2}}u~~\textrm{in}~{\mathbb {R}}^{2}, \end{array} \end{aligned}$$

(1.1)

where $a,b,\mu >0$, $2< q < p \le 6$. The problem (1.1) is related to the equation

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

(1.2)

proposed by Kirchhoff as an extension of the classical D’Alembert’s wave equations. Mathematically, the problem (1.2) is often referred to be nonlocal as the appearance of the term ${b\int _{{{\mathbb {R}}^N}} {|\nabla u{|^2}dx} }$ implies that (1.2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, which make the study of (1.2) particularly interesting.

After the pioneering work of Lions  [7], the Kirchhoff type problem (1.2) began to receive much attention and many important results were established, see  [1, 3, 9, 14]. When $f(x,u) = \lambda u + g(x,u)$, to find solutions for problem (1.2), a possible choice is to consider fixed $\lambda$, or even with an additional external and fixed potential V  [4, 6]. Another possible choice is to search for solutions to (1.2) having prescribed $L^2$-norm, see  [5, 12, 13]. In this case, (1.2) can be viewed as an eigenvalue problem by taking $\lambda$ as amn unknown Lagrange multiplier, and solved by studying some constrained variational problems. Inspired by the works of  [10, 12], we consider the following minimization problem

$$\begin{aligned} \begin{array}{ll} m(c): = \mathop {\inf }\limits _{{S_c}} {E_\mu }(u), \end{array} \end{aligned}$$

(1.3)

where ${E_\mu }(u) = \frac{a}{2}|\nabla u|_2^2 + \frac{b}{4}|\nabla u|_2^4 - \frac{1}{p}|u|_p^p - \frac{\mu }{q}|u|_q^q$, ${S_c} = \left\{ {u \in {H^1}({{\mathbb {R}}^2}):|u|_2^2 = c} \right\}$, and $| \cdot {|_r}$ denotes the standard norm in ${L^r}({{\mathbb {R}}^2})$. If $u \in {S_c}$ is a minimizer of problem (1.3), then there exists ${\lambda _c} \in {\mathbb {R}}$ such that ${E_\mu }'(u) = {\lambda _c}u$, namely, $u \in {S_c}$ is a solution of (1.1) for some $\lambda _c$. It is known that $p= 6$ is a $L^2$-critical exponent, that is, ${E_\mu }$ is bounded from below on $S_c$ if $2< p < 6$, while ${E_\mu }$ is not bounded from below on $S_c$ if $p > 6$.

When $\mu = 0$, Ye  [12] considered problem (1.3) with $L^2$-subcritical case, and proved the existence of minimizers to (1.3). Later, Zeng and Zhang  [13] gave a new proof for the results of  [12] by using technical energy estimates, and showed that the minimizer of (1.3), if exists, is unique and must be a scaling of Q, where Q is the unique radially symmetric positive solution of the following equation

$$\begin{aligned} \begin{array}{ll} - \Delta u + \frac{2}{{p - 2}}u - \frac{2}{{p - 2}}|u{|^{p - 2}}u = 0~~\textrm{in}~{\mathbb {R}}{^2}. \end{array} \end{aligned}$$

(1.4)

When $\mu \ne 0$, Li, Luo and Yang  [5] considered the existence of multiple solutions to problem (1.1) with the $L^2$-subcritical and supercritical case in ${\mathbb {R}}{^3}$ by constructing a suitable Palais-Smale sequence and Ekeland’s variational principle, and obtained the existence of ground state solution in the $L^2$-supercritical case by using minimax methods. However, to the best of our knowledge, the existence of normalized solutions to problem (1.1) with $L^2$-subcritical or critical case in ${\mathbb {R}}{^2}$ is still unknown.

Motivated by the above works, in this paper we consider that the existence of normalized solutions to problem (1.1) with $L^2$-subcritical or critical case, namely, $2< q < 4$ and $2 < p \le 6$ in ${\mathbb {R}}{^2}$. In addition, we discuss how $\mu$ affects the existence of normalized solutions to problem (1.1).

We recall that Q is an optimizer of the following sharp Gagliardo-Nirenberg  [11] inequality

$$\begin{aligned} \begin{array}{ll} |u\mathrm{{|}}_p^p \le \frac{p}{{2|Q\mathrm{{|}}_2^{p - 2}}}|\nabla u\mathrm{{|}}_2^{p-2}|u\mathrm{{|}}_2^2,~~\forall u \in {H^1}({\mathbb {R}}{^2}). \end{array} \end{aligned}$$

Moreover, combining the Pohozaev identity, Q satisfies $|\nabla Q|_2^2 = |Q|_2^2 = \frac{2}{p}|Q|_p^p$. Let

$$\begin{aligned} \begin{array}{ll} {c^*}:= {\frac{{b|Q|_2^4}}{2}},~~c_p^*(a,b):= {2|Q|_2^{p - 2}{{\left( {\frac{a}{{6-p}}} \right) }^{\frac{6-p}{2}}}{{\left( {\frac{b}{{2(p-4)}}} \right) }^{\frac{p-4}{2}}}}, \end{array} \end{aligned}$$

our first result is the nonexistence of minimizers to problem (1.3).

Fig. 1

The nonexistence (red area) and existence (green area) of constraint minimizers

Theorem 1.1

Assume that $2< q < p \le 6$ and c, $\mu > 0$. Then problem (1.3) has no minimizer if one of the following conditions holds.

• (i) $q = 4$, $p = 6$, either $c < {c^*}$ and $\mu c \le a|Q|_2^2$, or $c = {c^*}$ and $\mu c< a|Q|_2^2$.

• (ii) $q = 4< p < 6$, $\mu c < a|Q|_2^2$, and $c<c_p^*(a_0,b)$, where $a_0 = \frac{{a|Q|_2^2 - \mu c}}{{|Q|_2^2}} > 0$.

• (iii) $4< q < 6 = p$, $c < {c^*}$, and $\mu c < c_q^*(a,{b_0})$, where $b_0 = \frac{{b|Q|_2^4 - 2c}}{{|Q|_2^4}} > 0$.

Remark 1.1

(a) Theorem  1.1 may be illustrated by the red area shown in Figure  1. (b) When $\mu = 0$, problem (1.3) also has no minimizer under the conditions (i), (ii) and (iii) (see  [13]). Theorem  1.1 shows that the nonexistence of minimizers to problem (1.3) still holds for small $\mu$, which partially extends the result of Zeng and Zhang  [13]. (c) Theorem  1.1 shows that the nonexistence of minimizers to problem (1.3). In fact, we may proceed as in Theorem 1.2 of  [12] to prove that $E_\mu$ has no constraint critical point on $S_c$ under the conditions (i) by Gagliardo-Nirenberg inequality and Pohozaev identity. For the cases (ii) and (iii), we can further constrain $\mu$ and c by Young’s inequality such that $E_\mu$ has no constraint critical point on $S_c$.

Let $c_p^* = + \infty ~\textrm{if}~2< p < 6,~c_p^* ={c^*}~\textrm{if}~p = 6$, we have the following result.

Theorem 1.2

Assume that $2< q < 4$, $2< p \le 6$, $q<p$ and $\mu > 0$. Then the problem (1.3) has a minimizer for every $c \in (0,c_p^*)$.

Remark 1.2

(a) The results in Theorem  1.2 may be illustrated by the green area shown in Figure  1. (b) Theorem  1.2 shows the existence of normalized solutions to problem (1.1) with $2< q < 4$ and $2< p \le 6$, which partially extends existence result of Soave  [10] in case where $b = 0$ and $2< q< 4 \le p < +\infty$.

Remark 1.3

If $4 \le p \le 6$, Ye  [12] showed that the nonexistence of minimizers to (1.3) with $\mu = 0$ for small $c>0$. But Theorem  1.2 shows that $\mu > 0$ will affect the existence of minimizers to problem (1.3) for $c \in (0,c_p^*)$ in the case where $2< q < 4$ and $q<p$. The reason is that under the conditions of Theorem  1.2, we have $m(c)<0$, which guarantees that m(c) satisfies the strict sub-additivity inequalities in Lemma  2.4. This is the key to prove the existence of normalized solutions.

To prove Theorem  1.2, since that every minimizing sequence for (1.3) is bounded, we only need to exclude the two possibilities of vanishing and dichotomy, namely, ${u_n} \rightharpoonup 0~\textrm{in}~{H^1}({{\mathbb {R}}^2})$ and ${u_n} \rightharpoonup u \ne 0~\textrm{in}~{H^1}({{\mathbb {R}}^2}),~0< |u|_2^2 < c$. By vanishing lemma of Lions, the former can be avoided. The latter can be excluded by the strict inequality $m(c) < m(\alpha ) + m(c - \alpha )$ for every $0< \alpha < c$.

Regarding the notation, in this paper, we use $\rightarrow$ and $\rightharpoonup$ to denote the strong and weak convergence in the related function space respectively. $:=$ and $=:$ denote definitions. The rest of this paper is organized as follows. In Sect. 2, we present some preliminary results for Theorems  1.1 and  1.2. In Sect. 3, we prove Theorems  1.1 and  1.2.

2 Preliminary results for Theorems  1.1 and  1.2

Lemma 2.1

Assume $c,\mu >0$, then $E_\mu$ is coercive on $S_c$ if one of the following conditions holds.

• (i) $2< q< p < 6$.

• (ii) $2< q < 6 = p$, $c < {c^*}$.

• (iii) $q =4$, $p =6$, $c = {c^\mathrm{{*}}}$ and $\mu c < a|Q|_2^2$.

Especially, if the condition (iii) holds, then $E_\mu (u) > 0$ for every $u \in {S_c}$.

Proof

For every $u \in {S_c}$, we have $E_\mu (u) \ge \frac{a}{2}|\nabla u|_2^2 + \frac{b}{4}|\nabla u|_2^4 - \frac{c}{{2|Q|_2^{p - 2}}}|\nabla u|_2^{p-2} - \frac{\mu c}{{2|Q|_2^{q - 2}}}|\nabla u|_2^{q-2}$. It is easy to check that $E_\mu (u) \rightarrow +\infty$ as $|\nabla u{|_2} \rightarrow +\infty$. $\square$

Lemma 2.2

• (1) Assume that the conditions of Lemma  2.1 hold, then m(c) is well defined and $m(c) \le 0$.

• (2) Assume that $2< q < 4$, $2< p \le 6$, $q<p$ and $\mu > 0$, then $m(c)<0$ for every $c \in (0,c_p^*)$.

Proof

• (1) By Lemma  2.1, $E_\mu$ is bounded from below on $S_c$. Let ${u_t}(x) = {t^{\frac{1}{2}}}u\left( {{t^{\frac{1}{2}}}x} \right)$, then ${u_t} \in {S_c}$, and $E_\mu ({u_t}) = \frac{a}{2}t|\nabla u|_2^2 + \frac{b}{4}{t^2}|\nabla u|_2^4 - \frac{1}{p}{t^{\frac{p-2}{2}}}|u|_p^p - \frac{\mu }{q}{t^{\frac{q-2}{2}}}|u|_q^q \rightarrow 0$ as $t \rightarrow 0$. Thus $m(c) \le 0$.

• (2) Let ${u_s}(x) = {c^{\frac{1}{2}}}\frac{{{s^{\frac{1}{2}}}Q\left( {{s^{\frac{1}{2}}}x} \right) }}{{|Q{|_2}}}$ for $s > 0$, then ${u_s} \in {S_c}$, and

  $$\begin{aligned} \begin{array}{ll} E_\mu ({u_s})= \frac{a}{2}cs + \frac{b}{4}{c^2}{s^2} - \frac{{{c^{\frac{p}{2}}}}}{{2|Q|_2^{p - 2}}}{s^{\frac{p-2}{2}}} - \frac{{\mu {c^{\frac{q}{2}}}|Q|_q^q}}{{q|Q|_2^q}}{s^{\frac{q-2}{2}}}. \end{array} \end{aligned}$$

  It is easy to see that there exists ${s_0} > 0$ such that $E_\mu ({u_{{s_0}}}) < 0$ when $2< q < 4$, $2 < p \le 6$ and $c \in (0,c_p^*)$. $\square$

Remark 2.1

By Lemma  2.1, we know that if $q = 4$, $p =6$, $c = {c^\mathrm{{*}}}$ and $\mu c < a|Q|_2^2$, then $E_\mu (u) > 0$ for every $u \in {S_c}$, so m(c) has no minimizer.

Lemma 2.3

Assume that $2< q < 4$, $2< p \le 6$, $q<p$ and $\mu > 0$. Then m(c) is continuous on $(0, c_p^*)$.

Proof

The proof is similar to that of Theorem 2.1 in  [2], so we omit it.

Lemma 2.4

Assume that $2< q <4$, $2< p \le 6$, $q<p$ and $\mu > 0$. Then for every $c \in (0,c_p^*)$, $m(c) < m(\alpha ) + m(c - \alpha )$ when $0< \alpha < c$.

Proof

By Lemma  2.2, $m(c)<0$ for every $c \in (0,c_p^*)$. Let $\{ {u_n}\} \subset {S_c}$ be a minimizing sequence for m(c), that is, $E_\mu ({u_n}) \rightarrow m(c)$. Since $E_\mu$ is coercive on $S_c$, then $\{ {u_n}\}$ is bounded in ${H^1}({{\mathbb {R}}^2})$. We claim that there exists a constant ${k_1} > 0$ such that $|\nabla {u_n}|_2^2 \ge {k_1}$. In fact, if $|\nabla {u_n}{|_2} \rightarrow 0$ then

$$\begin{aligned} \begin{array}{ll} E_\mu ({u_n}) \ge \frac{a}{2}|\nabla {u_n}|_2^2 + \frac{b}{4}|\nabla {u_n}|_2^4 - \frac{c}{{2|Q|_2^{p - 2}}}|\nabla {u_n}|_2^{p-2} - \frac{\mu c}{{2|Q|_2^{q - 2}}}|\nabla {u_n}|_2^{q-2} \rightarrow 0. \end{array} \end{aligned}$$

Since $E_\mu ({u_n}) \le \frac{a}{2}|\nabla {u_n}|_2^2 + \frac{b}{4}|\nabla {u_n}|_2^4 \rightarrow 0$, then $E_\mu ({u_n}) \rightarrow 0$, a contradiction.

For every $u \in {S_c}$, let ${u^\theta }(x) = u\left( {{\theta ^{ - \frac{1}{2}}}x} \right)$ for $\theta > 0$, then ${u^\theta } \in {S_{\theta c}}$, so $u_n^\theta \in {S_{\theta c}}$. Let $\theta > 1$ such that $\theta c < c_p^*$, then

$$\begin{aligned} \begin{array}{ll} m(\theta c) &{}\le {E_\mu }(u_n^\theta ) = \frac{a}{2}|\nabla {u_n}|_2^2 + \frac{b}{4}|\nabla {u_n}|_2^4 - \frac{1}{p}\theta |{u_n}|_p^p - \frac{\mu }{q}\theta |{u_n}|_q^q\\ &{} = \theta {E_\mu }({u_n}) + \frac{a}{2}\left( {1 - \theta } \right) |\nabla {u_n}|_2^2 + \frac{b}{4}\left( {1 - \theta } \right) |\nabla {u_n}|_2^4\\ &{} \le \theta {E_\mu }({u_n}) + \frac{a}{2}\left( {1 - \theta } \right) {k_1} + \frac{b}{4}\left( {1 - \theta } \right) k_1^2. \end{array} \end{aligned}$$

Since the second term on the right is negative, we have $m(\theta c) < \theta m(c)$, and from this the thesis follows as in Lemma II.1 of  [8]. $\square$

3 Proof of Theorems  1.1 and  1.2

Proof of Theorem 1.1

It is enough to show that $E_\mu (u) > 0$ for every $u \in {S_c}$. Since $m(c) \le 0$, then m(c) has no minimizer.

• (i) If $q = 4$, $p = 6$, and either $c < {c^*}$, $\mu c \le a|Q|_2^2$ or $c = {c^*}$, $\mu c < a|Q|_2^2$, then by Gagliardo-Nirenberg inequality, we have $E_\mu (u) \ge \frac{1}{2}\left( {a - \frac{\mu c}{|Q|_2^2}} \right) |\nabla u|_2^2 + \frac{1}{2}\left( {\frac{b}{2} - \frac{c}{|Q|_2^4}} \right) |\nabla u|_2^4 > 0$.

• (ii) If $q = 4< p <6$ and $\mu c < a|Q|_2^2$, then

  $$\begin{aligned} \begin{array}{ll} E_\mu (u) \ge \frac{1}{2}\left( {a - \frac{\mu c}{|Q|_2^2}} \right) |\nabla u|_2^2 + \frac{b}{4}|\nabla u|_2^4 - \frac{c}{{2|Q|_2^{p - 2}}}|\nabla u|_2^{p-2}. \end{array} \end{aligned}$$

  Let $|\nabla u|_2^2 = t,~\alpha = \frac{{6 - p}}{2},~\beta = \frac{{p - 4}}{2}$. By Young’s inequality, we deduce that

  $$\begin{aligned} \frac{a_0}{2}t + \frac{b}{4}{t^2} = \alpha \cdot \frac{a_0}{{2\alpha }}t + \beta \cdot \frac{b}{{4\beta }}{t^2} \ge {\left( {\frac{a_0}{{2\alpha }}} \right) ^\alpha }{\left( {\frac{b}{{4\beta }}} \right) ^\beta }{t^{\alpha + 2\beta }} = {\left( {\frac{a_0}{{2\alpha }}} \right) ^\alpha }{\left( {\frac{b}{{4\beta }}} \right) ^\beta }{t^{\frac{p-2}{2}}}, \end{aligned}$$

  where the equality holds if and only if $t = \frac{{2(p-4)a_0}}{{(6-p)b}}$. Thus $E_\mu (u) \ge \frac{c_p^*(a_0,b) - c}{2|Q|_2^{p - 2}}|\nabla u|_2^{p-2}>0$ for every $u \in {S_c}$.

• (iii) If $4< q < 6 = p$, $c < {c^*}$ and $\mu c < c_q^*(a,{b_0})$, then for every $u \in {S_c}$, by Young’s inequality

  $$\begin{aligned} \begin{array}{ll} E_\mu (u) \ge \frac{a}{2}|\nabla u|_2^2 + \frac{1}{4}\left( {b - \frac{2c}{|Q|_2^4}} \right) |\nabla u|_2^4 - \frac{{\mu c}}{{2|Q|_2^{q - 2}}}|\nabla u|_2^{q-2} \ge \frac{c_q^*(a,b_0) - \mu c}{2|Q|_2^{q - 2}}|\nabla u|_2^{q-2} >0. \end{array} \end{aligned}$$

Proof of Theorem 1.2

Let $\{ {u_n}\} \subset {S_c}$ be a minimizing sequence for m(c), namely, $E_\mu ({u_n}) \rightarrow m(c)$ and $|{u_n}|_2^2 = c$. Since $E_\mu$ is coercive on ${S_c}$, then $\{ {u_n}\}$ is bounded in ${H^1}({{\mathbb {R}}^2})$. Then

$$\begin{aligned} \begin{array}{ll} \delta : = \mathop {\underline{\lim } }\limits _{n \rightarrow \infty } \mathop {\sup }\limits _{y \in {{\mathbb {R}}^2}} \int _{{B_1}(y)} {|{u_n}{|^2}dx} > 0. \end{array} \end{aligned}$$

Otherwise, by vanishing lemma of Lions, ${u_n} \rightarrow 0$ in ${L^r}({{\mathbb {R}}^2})$ for $2< r < +\infty$, and then $0 \le \mathop {\lim }\limits _{n \rightarrow \infty } \left( {\frac{a}{2}|\nabla {u_n}|_2^2 + \frac{b}{4}|\nabla {u_n}|_2^4} \right) = \mathop {\lim }\limits _{n \rightarrow \infty } E_\mu ({u_n}) = m(c) < 0$, a contradiction. Therefore, there exists a sequence $\{ {y_n}\} \subset {{\mathbb {R}}^2}$ such that $\int _{{B_1}({y_n})} {|{u_n}{|^2}dx}> \frac{\delta }{2} > 0$. Let ${v_n} = {u_n}(x + {y_n})$, and then

$$\begin{aligned} \begin{array}{ll} \int _{{B_1}(0)} {|{v_n}{|^2}dx} > \frac{\delta }{2}. \end{array} \end{aligned}$$

(3.1)

Moreover $\{ {v_n}\} \subset {S_c}$ also a bounded minimizing sequence for m(c), and then we may assume that ${v_n} \rightharpoonup {v_0}~\textrm{in}~{H^1}({{\mathbb {R}}^2})$, ${v_n} \rightarrow {v_0}$ in $L_{loc}^ r({{\mathbb {R}} ^2})$ for $r \in [1,+\infty )$, ${v_n}(x) \rightarrow {v_0}(x)$ a.e. in ${{\mathbb {R}}^2}$, which and (3.1) imply that ${v_0} \ne 0$. Thus $\alpha : = |{v_0}|_2^2 \in (0,c]$.

We now prove that $\alpha = c$. Suppose that $\alpha < c$, then $c = |{v_n}|_2^2 = |{v_0}|_2^2 + |{v_n} - {v_0}|_2^2 + o(1)$. Combining Brezis-Lieb Lemma and Lemma  2.3, we see that

$$\begin{aligned} m(c) = \mathop {\lim }\limits _{n \rightarrow \infty } E_\mu ({v_n}) = E_\mu ({v_0}) + \mathop {\lim }\limits _{n \rightarrow \infty } E_\mu ({v_n} - {v_0}) \ge m(\alpha ) + m(c - \alpha ), \end{aligned}$$

which contradicts Lemma  2.4. So $\alpha = c$, namely, ${v_0} \in {S_c}$. Since $m(c) \le E_\mu ({v_0}) \le \mathop {\underline{\lim } }\limits _{n \rightarrow \infty } E_\mu ({v_n}) = m(c)$, then ${v_0} \in {S_c}$ is a minimizer of m(c). $\square$