Existence of normalized solutions for a Sobolev supercritical Schrödinger equation

1.   Introduction and main results

In this paper, we study the existence of $L^2$-normalized solutions to the following equation:

where $p > 2^*: = \frac{2N}{N-2}$, $N\geq 3$, $f: \mathbb{R} \rightarrow \mathbb{R}$ is a nonlinear function, that is not necessarily of the power type, and the mass $a > 0$ is given, while a pair $(\lambda, u) \in \mathbb{R} \times H^1\left(\mathbb{R}^N\right)$ is unknown.

Problem (1.1) appears when searching for standing waves to the following nonlinear Schrödinger equation

i.e. solutions to (1.2) of the form $\psi(t, x) = e^{i \lambda t} u(x)$ for some $\lambda \in \mathbb{R}$. The constraint $\int_{\mathbb{R}^N}|u|^2dx = a^2$ comes from the fact that, under suitable assumptions on $f$, a solution $\psi$ of (1.2) preserves such a quantity in time $\int_{\mathbb{R}^N}|\psi(t, x)|^2 d x = \int_{\mathbb{R}^N}|\psi(0, x)|^2 d x$ for all $t \in(0, T)$.

To date, two distinct perspectives have emerged regarding the frequency $\lambda$ in Eq (1.1). One approach views $\lambda$ as a predetermined constant, leading to the so-called fixed frequency problem. Extensive research has been conducted on the existence, multiplicity, and concentration of nontrivial solutions to (1.1) under various assumptions on the nonlinearity $f$ and the potential $V$, encompassing Sobolev subcritical, critical, and supercritical growth regimes (see ^[1,2,3]).

Alternatively, $\lambda$ can be considered an unknown quantity within Eq (1.1). In this context, it is natural to prescribe the value of the mass, allowing $\lambda$ to be interpreted as a Lagrange multiplier. Notably, the mass possesses a clear physical significance. For instance, in Bose-Einstein condensates, $\lambda$ corresponds to the chemical potential, representing the energy required to add one particle to the system. In nonlinear optics, $\lambda$ reflects the refractive index shift induced by nonlinear effects. Thus, $\lambda$ is not merely a Lagrange multiplier in the mathematical model but is also closely tied to the system's physical properties, offering insights into its dynamics and characteristics.

Furthermore, such solutions offer valuable insights into the dynamical properties of the system, including orbital stability or instability, and can effectively describe attractive Bose-Einstein condensates. In the mathematical literature, these types of solutions are commonly referred to as prescribed $L^2$-norm solutions or normalized solutions. We refer to ^[4,5,6] and references therein.

In recent decades, the existence of $L^2$-normalized solutions has been a topic of active research, resulting in a substantial body of literature. The $L^2$-critical exponent $p$, defined as

plays a key role in analyzing (1.1) and the stability properties of (1.2), as discussed in ^[7]. Note that $p \in (2, 2^*)$, where, as usual, $2^* = \frac{2N}{N-2}$ for $N \geq 3$, and $2^* = \infty$ for $N = 2$.

To illustrate the importance of the $L^2$-critical exponent $p$, consider $q \in (2, 2^*)$ and $\lambda > 0$. Recall that the equation

has a unique positive solution up to translations (see Kwong ^[8]). Its unique positive radial solution is denoted by $Q_{q, \lambda}$. It is easily observed that

where $Q_{q, 1}$ is the unique positive radial solution of

Setting the mass

and the energy

it follows from the Pohozaev identity that

and

Thus we see that $p = 2+\frac{4}{N}$ plays a special role. Indeed, considering three cases

then the energy $\mathcal{E}_q(u)$ of a solution $(\lambda, u)$ for (1.1) satisfies

Recent studies have concentrated on normalized solutions of the Schrödinger equation, with particular attention to the Sobolev subcritical case. Soave ^[9] examined the existence and properties of ground states for the nonlinear Schrödinger equation involving combined power nonlinearities:

on

where $N \geq 1$, $2 < q\leq 2+\frac{4}{N}\leq p < 2^*$. Yang et al. ^[10] obtained the existence and multiplicity of normalized solutions to the following Schrödinger equations with potentials and non-autonomous nonlinearities:

where $f(x, s)$ satisfies Berestycki-Lions type conditions with mass subcritical growth. Claudianor, Alves and Thin ^[11,12] investigated the existence of multiple normalized solutions to the following class of elliptic problems:

under different assumptions about potentials, but $f$ verifies weak mass subcritical growth. For Sobolev critical case, Soave ^[13] considered the Sobolev critical problem

where $2+\frac{4}{N} < q < 2^*$, $N \geq 3$, but the upper boundedness of $\mu\cdot a^{(1-\gamma_q)q}$ is essential. Li and Zou ^[14] obtained the existence of ground states for (1.4) that does not depend on the range of $\mu\cdot a^{(1-\gamma_q)q}$, which improves and extends the result in ^[13]. Bieganowski and Mederski ^[15] proposed a simple minimization method based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints to prove the existence of normalized ground states to

for the Sobolev subcritical equation.

Existing research predominantly concentrates on normalized solutions within the context of Sobolev subcritical or critical problems. To the best of our knowledge, the Sobolev supercritical case has not been explored in relation to normalized solutions. In this paper, we delve into this intriguing subject and introduce the following assumptions:

$(V)$ $V \in C(\mathbb{R}^N, \mathbb{R})$, $0 < V_0: = \inf\limits_{x \in \mathbb{R}^N}V(x) \leq V(x) \leq V_{\infty}: = \lim\limits_{|x|\rightarrow \infty}V(x) < +\infty$ for all $x \in \mathbb{R}^N$.

$(f_1)$ $f \in C(\mathbb{R}, \mathbb{R})$ is odd, and there exist $m \in (2, 2+\frac{4}{N})$ and $\alpha \in (0, +\infty)$ such that

$(f_2)$ There exist two constants $C_1, \ C_2 > 0$ and $q \in (m, 2+\frac{4}{N})$ such that

$(f_3)$ $\frac{f(t)}{t^{q-1}}$ is an increasing function of $t$ on $(0, +\infty)$.

Theorem 1.1. Suppose that $(V)$ and $(f_1)$-$(f_3)$ are satisfied. Then there exists some $\mu_0 > 0$ such that for $\mu \in (0, \mu_0]$, problem (1.1) admits a couple of weak solutions $(u, \lambda) \in H^1(\mathbb{R}^N) \times \mathbb{R}^+$ with $\int_{\mathbb{R}^N}|u|^2dx = a^2$.

2.   Some lemmas and proof of Theorem 1.1

Define a function

where $M > 0$. Then

and

Set $h_\mu(t) = \mu\phi(t)+f(t)$ for all $t \in R$. Then $h_\mu(t)$ possesses the following properties:

$(h_1) h_\mu \in C(\mathbb{R}, \mathbb{R})$ is odd, and $\lim\limits_{t\rightarrow 0}\frac{|h_\mu(t)|}{|t|^{m-1}} = \alpha$.

$(h_2)$ $|h_\mu(t)|\leq \mu M^{p-q}|t|^{q-1}+C_1+C_2|t|^{q-1}$ for all $t \in \mathbb{R}$.

$(h_3)$ $\frac{h_\mu(t)}{t^{q-1}}$ is an increasing function of $t$ on $(0, +\infty)$.

$(h_4)$ $h_\mu(t)t\geq qH_\mu(t): = q\int_0^th_\mu(s)ds\geq0$ for all $t \in \mathbb{R}$.

By $(V)$ and $(h_1)$-$(h_3)$ and ^[11], the following problem

admits a couple $(u_\mu, \lambda) \in H^1(\mathbb{R}^N) \times \mathbb{R}^+$ of weak solutions with $\int_{\mathbb{R}^N}|u_\mu|^2dx = a^2$. Let

Then

and

where

Consequently, there exists $\lambda \in \mathbb{R}$ such that

Moreover, set

and

where

Then, $\gamma_{0, a, \mu}\leq \gamma_{0, a}$.

Lemma 2.1. The solution $u_\mu$ satisfies $\|\nabla u_\mu\|_2^2\leq \frac{2(q\gamma_{0, a}+\lambda a^2)}{q-2}$.

Proof. By (2.1), we have

Therefore,

Which implies that the lemma holds. □

Lemma 2.2. There exist two constants $B, \ D > 0$ independent of $\mu$ such that $\|u_\mu\|_{L^{\infty}}\leq B(1+\mu)^D$.

Proof. Let $b: \mathbb{R}\rightarrow \mathbb{R}$ be a smooth function satisfying:

● $b(s) = s$ for $|s|\leq T-1$;

● $b(-s) = -b(s)$;

● $b'(s) = 0$ for $s\geq T$;

● $b'(s)$ is decreasing in $[T-1, T]$.

Set $T > 2$, $r > 0$, and $\tilde{u}_\mu^{T}£º = b(u_\mu)$. It is easy to see that

where $T-1\leq C_T\leq T$. Moreover, $0\leq \frac{sb^{\prime}(s)}{b(s)}\leq 1, \forall \ s\neq 0$. Let $\psi = u_\mu|\tilde{u}_\mu^{T}|^{2r}$. Then $\psi \in H^1(\mathbb{R}^N)$. Taking $\psi$ as the test function, there holds

As a consequence, from $(h_1)$ and $(h_2)$, for fixed $\mu > 0$ and small $\varepsilon > 0$,

Combining with the above equations, we obtain

By the Sobolev embedding theorem,

As a result,

Take $r_0 > 0$ and $r_{_k} = r_0(\frac{2^*}{2})^k = r_{_{k-1}}\cdot\frac{2^*}{2}$. Then

Notice that

Take $\rho > 0$ be such that $C(r_0+1)^2(\int\limits_{|u_\mu(x)|\geq \rho}|u_\mu|^{2^*}dx)^{\frac{2}{N}} < \frac{1}{2}$. Then

Set

and

Then $d_k\rightarrow d_{\infty}$ as $k\rightarrow \infty$ and $e_k\rightarrow e_{\infty} = (1+\mu)^{\frac{2^*}{(2^*-2)2r_0}}$ as $k\rightarrow \infty$. By Lemma 2.1, we have

Using Fatous lemma in $T\rightarrow +\infty$, we obtain

Consequently, let $k\rightarrow \infty$, we obtain

This completes the proof. □

Proof of Theorem 1.1: For large $M > 0$, we can choose small $\mu_0 > 0$ such that $\|u_\mu\|_{L^{\infty}}\leq B(1+\mu)^D\leq M$ for all $\mu \in (0, \mu_0]$. Which indicates $h_\mu(u_\mu) = \mu|u_\mu|^{p-2}u_\mu+f(u_\mu)$. Consequently, problem (1.1) admits a couple of weak solutions $(u_\mu, \lambda) \in H^1(\mathbb{R}^N) \times \mathbb{R}^+$ with $\int_{\mathbb{R}^N}|u_\mu|^2dx = a^2$. This completes the proof.

3.   Conclusions

In this paper, we study the existence of normalized solutions for the following Schrödinger equation with Sobolev supercritical growth:

where $p > 2^*: = \frac{2N}{N-2}$, $N\geq 3$, $a > 0$, $\lambda \in \mathbb{R}$ is an unknown Lagrange multiplier, $V \in C(\mathbb{R}^N, \mathbb{R})$, $f$ satisfies the following assumptions:

$(V)$ $V \in C(\mathbb{R}^N, \mathbb{R})$, $0 < V_0: = \inf\limits_{x \in \mathbb{R}^N}V(x) \leq V(x) \leq V_{\infty}: = \lim\limits_{|x|\rightarrow \infty}V(x) < +\infty$ for all $x \in \mathbb{R}^N$.

$(f_1)$ $f \in C(\mathbb{R}, \mathbb{R})$ is odd, and there exist $m \in (2, 2+\frac{4}{N})$ and $\alpha \in (0, +\infty)$ such that

$(f_2)$ There exist two constants $C_1, \ C_2 > 0$ and $q \in (m, 2+\frac{4}{N})$ such that

$(f_3)$ $\frac{f(t)}{t^{q-1}}$ is an increasing function of $t$ on $(0, +\infty)$.

By employing the truncation technique, we establish the existence of normalized solutions to this Sobolev supercritical problem. In particular, there exists some $\mu_0 > 0$ such that for $\mu \in (0, \mu_0]$, problem (3.1) admits a couple of weak solutions $(u, \lambda) \in H^1(\mathbb{R}^N) \times \mathbb{R}^+$ with $\int_{\mathbb{R}^N}|u|^2dx = a^2$.

Use of AI tools declaration

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

Acknowledgments

Q. Li is supported in part by the National Natural Science Foundation of China (12261031), Yunnan Province Applied Basic Research for Key Project (202401AS070024), Yunnan Province Applied Basic Research for General Project (202301AT070141), and Youth Outstanding-notch Talent Support Program in Yunnan Province and the Project Funds of Xingdian Talent Support Program. Z. Yang is supported by the Xingdian talent support program of Yunnan Province, the National Natural Science Foundation of China (12301145, 12261107), Yunnan Province Applied Basic Research for Youths Projects (202301AU070144), the Yunnan Province Applied Basic Research for General Project (202401AU070123), the Scientific Research Fund of Yunnan Educational Commission (2023J0199), and the Yunnan Key Laboratory of Modern Analytical Mathematics and Applications (202302AN360007).

Conflict of interest

The authors declare there is no conflicts of interest.