Existence of normalized solutions for the Schrödinger equation

1.   Introduction

In this paper, we consider the existence of solutions for the following Schrödinger equation.

where $N\geqslant 3$, $2 < q < 2+\frac{4}{N}$ and $p = 2^* = \frac{2N}{N-2}$. The Schrödinger equation is a famous equation in Physics and there are numerous papers to study it, we refer the readers to ^[1,2,3,4] and references therein.

For (1.1), we are particularly interested in the stationary waves of the form $\psi(x, t) = e^{-i\lambda t}u(x)$, where $\lambda\in\mathbb{R}$ and $u:\mathbb{R}^N\rightarrow \mathbb{R}$. Then $u$ satisfies the equation

If we fix the $L^2$-norm of $u$, that is, let

where $a > 0$ is a constant. Then the corresponding functional of (1.2) is

and $\lambda$ appears as a Lagrange multiplier. Solutions of (1.2) with prescribed mass are always called normalized solutions. It seems that there is profound physical significance to study normalized solutions. In fact, for the Schrödinger equation, $\lvert\psi(x, t)\rvert^2$ represents the probability density of a single particle appearing in space $x$ at time $t$. Naturally, there is

Of course, in mathematics, we often consider

There are a lot of papers to study the normalized solutions of Schrödinger equations and it is impossible for us to provide complete references. We refer the readers to ^{[5,6,7,8,9,10,11]} and references therein. Moreover, we refer the readers to ^[12,13,14] for the normalized solutions of fractional Schrödinger equations and to ^[15,16,17] for the normalized solutions of Schrödinger systems.

When we study the normalized solutions, there will be a $L^2$-critical exponent $2+\frac{4}{N}$, which comes from the Gagliardo-Nirenberg inequality^[18]: for every $2 < p < 2^*$, there exists an optimal constant $C_{N, p}$ depending on $N$ and $p$ such that

where

By the Gagliardo-Nirenberg inequality, it is not difficult to prove that if the nonlinearities of equation are $L^2$-subcritical, then the corresponding functional is bounded from below on $S_{a}$. For example,

is bounded from below on $S_{a}$ for $2 < p < 2+\frac{4}{N}$ and global minimizers of $J|_{S_{a}}$ can be found, see ^[8,19]. However, if the nonlinearities are $L^2$-supercritical, the functional is unbounded from below on $S_{a}$ and it seems impossible to search for a global minimizer. The first paper to deal with $L^2$-supercritical is ^[5]. In ^[5], Jeanjean found the normalized solutions of mountain-pass type.

Compared with pure $L^2$-subcritical or $L^2$-supercritical case, the mixed case is more complicated. In ^[9], Soave studied (1.2) for $2 < q < 2+\frac{4}{N} < p < 2^*$ under $L^2$ constraint. Since $q$ is $L^2$-subcritical exponent and $p$ is $L^2$-supercritical exponent, we call $\mu\lvert u\rvert^{q-2}+\lvert u\rvert^{p-2}u$ mixed nonlinearities. The first existence result of normalized solutions in Sobolev critical case was also obtained by Soave^[10].

Since the $L^2$ constraint, there are some difficulties to observe the structure of $E_{p}|_{S_{a}}$. A possible method is to consider the function

where

It is not difficult to prove that $s\star u\in S_{a}$ for all $s\in\mathbb{R}$ if $u\in S_{a}$ and hence we can study the structure of $\Psi_{u}^{p}$ to speculate the structure of $E_{p}|_{S_{a}}$.

If $u$ is a critical point of $E_{p}|_{S_{a}}$, then $0$ may be a critical point of $\Psi_{u}^{p}$. If $0$ is a critical point of $\Psi_{u}^{p}$, then $(\Psi_{u}^{p})'(0) = 0$, that is

In fact, by Pohozaev identity, $u$ satisfies (1.3) as long as $u$ is a critical point of $E_{p}$. Now, we can define a manifold

where

It is clear that all critical points of $E_{p}|_{S_{a}}$ belong to $\mathcal{P}_{a, p}$ and $s\star u\in\mathcal{P}_{a, p}$ if and only if $(\Psi_{u}^{p}(s))' = 0$. We divide $\mathcal{P}_{a, p}$ into three parts.

and

Define

For $2 < q < 2+\frac{4}{N} < p\leqslant 2^*$, since $q\gamma_{q} < 2$ and $p\gamma_{p} > 2$, the function $\Psi_{u}^{p}$ may have two critical points on $\mathbb{R}$, one is local minimum point and the other is global maximum point. Moreover, if we assume $s_{u}$ is the local minimum and $t_{u}$ is the global maximum. Then, it is not difficulty to check that $s_{u}\star u\in\mathcal{P}_{a, p}^{+}$ and $t_{u}\star u\in\mathcal{P}_{a, p}^{-}$ (see [9, Lemma 5.3] and [10, Lemma 4.2] for more details). Therefore, it is natural to speculate that $E_{p}$ has two critical points on $S_{a}$ under appropriate assumptions, one is a local minimizer on $S_{a}$ and is also a minimizer on $\mathcal{P}_{a, p}^{+}$, the other is a mountain-pass type critical point and is also a minimizer on $\mathcal{P}_{a, p}^{-}$.

In fact, the local minimizer and mountain-pass type solution of $E_{p}|_{S_{a}}$ for $2 < q < 2+\frac{4}{N} < p < 2^*$ have been found by Soave, see [9, Theorem 1.3]. For $2 < q < 2+\frac{4}{N} < p = 2^*$, Soave obtained the local minimum, but due to $H_{rad}^1(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$ is not compact, there are some difficulties to obtain the mountain-pass type solution (see Theorem 1.1 and Remark 1.1 in ^[10]). Therefore, it is natural to ask the following question:

(Q) Does $E_{2^*}|_{S_{a}}$ has a second critical point of mountain pass type when $2 < q < 2+\frac{4}{N}$? In ^[6], Jeanjean and Le proved $E_{2^*}|_{S_{a}}$ has a mountain-pass type solution and the solution is also a minimizer on $\mathcal{P}_{a, 2^*}^{-}$ when $N\geqslant 4$. They constructed a minimax structure and proved a strict inequality $m^{-}(a, 2^*) < m^{+}(a, 2^*)+\frac{1}{N}S^{\frac{N}{2}}$ to obtain the compactness of a Palais-smale(PS) sequence. The proof of ^[6] is complicated especially the proof of the strict inequality, see Propositions 1.10, 1.11 and 1.12 for more details. After that, Wei and Wu ^[11] gave a simpler proof of $m^{-}(a, 2^*) < m^{+}(a, 2^*)+\frac{1}{N}S^{\frac{N}{2}}$ and proved that the answer is also positive for (Q) when $N = 3$. Different from ^[6], We and Wu didn't construct the minimax structure, but directly proved the convergence of the minimizing sequence for $m^{-}(a, 2^*)$, see Lemma 3.1 and Proposition 3.1 of ^[11] for more details.

Our main goal is giving a new proof of (Q) and the method we call the Sobolev subcritical approximation method. The idea of the Sobolev subcritical approximation method is: by [9, Theorem1.3 (ii)], we know $E_{p}|_{S_{a}}$ has a mountain-pass type solution $u_{p}$ when $2+\frac{4}{N} < p < 2^*$. Let $p\rightarrow 2^*$, it is not difficult to prove that $u_{p}\rightharpoonup u$ in $H^1(\mathbb{R}^N)$. Then, we prove that $u$ is the solution of (1.2), $u_{p}\rightarrow u$ in $H^1(\mathbb{R}^N)$, $u$ is a critical point of $E_{2^*}|_{S_{a}}$ and is the minimum of $E_{2^*}$ on $\mathcal{P}_{a, 2^*}^{-}$. Proving strong convergence is a crucial step in our proof, we also need use the strict inequality $m^{-}(a, 2^*) < m^{+}(a, 2^*)+\frac{1}{N}S^{\frac{N}{2}}$.

Let

and

Define $\alpha(N, q): = \min\{C', C''\}$. Our main result can be stated as follows.

Theorem 1.1. Let $N\geqslant 3$, $2 < q < 2+\frac{4}{N}$, $p = 2^*$ and $a, \mu > 0$. Moreover, let us suppose that $\mu a^{q(1-\gamma_{q})} < \alpha(N, q)$. Then $E_{2^*}|_{S_{a}}$ has a critical point of mountain-pass type which is positive, radially symmetric and solves (1.2) for some $\lambda<0$.

Remark 1.1. The definition of $\alpha(N,q)$ comes from [10, (1.6)] to ensure that $\Psi_{u}^{p}$ has two critical points.

Remark 1.2. The Sobolev subcritical approximation method has been used by [20, Remark 1.3] and ^[7]. In ^[7], Li considered the normalized solutions of (1.2) with $2+\frac{4}{N}<q<p = 2^*$ and proved (1.2) has a normalized ground state for every $\mu>0$, see [7, Theorem 1.4]. Li solve an open problem

(Q') Does $E_{2^*}|_{S_{a}}$ have a ground state if $\mu>0$ and $\mu a^{(1-\gamma_{q})q}$ large? which is raised by Soave [10, Page 7]. For $2<q<2+\frac{4}{N}<p = 2^*$, if we follow the step of Li, the last inequality is invalid since $q\gamma_{q}<2$ (see [7, Page 13]) and we can not prove $u\in S_{a}$. In fact, we refer some ideas of ^[10,11] to obtain strong convergence in $H^1(\mathbb{R}^N)$.

2.   Preliminaries

In this section, we collect some results which will be used in the rest of the paper. First, let us recall the Sobolev inequality.

Lemma 2.1. For every $N\geqslant 3$, there is an optimal constant $S>0$ depending only on $N$ such that

where $D^{1, 2}(\mathbb{R}^N)$ denotes the completion of $C_{c}^{\infty}(\mathbb{R}^N)$ with respect to the norm $\lVert u\rVert_{D^{1, 2}}: = \lVert\nabla u\rVert_{2}$.

It is well known ^[21] that $S$ is achieved by

and $U_{\varepsilon, y}$ satisfies the equation

Moreover,

Let $C_{N, p}$ be the optimal constant of Gagliardo-Nirenberg inequality. Then, we have

Lemma 2.2. Let $2 < p < 2^*$, then $\lim_{p\rightarrow 2^*}C_{N, p} = S^{-\frac{1}{2}}$.

Proof. Denoting by $u_{\varepsilon}: = \varphi U_{\varepsilon, 0}\in H^1(\mathbb{R}^N)$, where $\varphi\in C_{c}^{\infty}(\mathbb{R}^N)$ be a radial cut-off function with

By the classical results^[20], we have

Since

the Lebesgue dominated convergence theorem implies $\lim_{p\rightarrow 2^*}\lVert u_{\varepsilon}\rVert_{p}^p = \lVert u_{\varepsilon}\rVert_{2^*}^{2^*}$. Using the Gagliardo-Nirenberg inequality, we have

Taking $p\rightarrow 2^*$, we obtain

which implies $S^{-\frac{1}{2}}\leqslant\liminf_{p\rightarrow 2^*}C_{N, p}.$

For every $u\in H^1(\mathbb{R}^N)\backslash\{0\}$, using the Hölder inequality and the Sobolev inequality, we have

By the definition of $C_{N, p}$, we obtain $S^{-\frac{\gamma_{p}}{2}}\geqslant C_{N, p}$. Therefore, $S^{-\frac{1}{2}}\geqslant\limsup_{p\rightarrow 2^*}C_{N, p}$.

3.   Proof of Theorem 1.1

For every $0 < \mu < a^{q(\gamma_{q}-1)}\alpha(N, q)$. In order to use the existence result of Sobolev subcritical case[9, Theorem 1.3], $\mu$ should satisfy

By Lemma 2.2, is it not difficult to prove that

as $p\rightarrow 2^*$, where $C'$ is defined by (1.4). Therefore, $\mu$ satisfies (3.1) as long as $p$ is close enough to $2^*$.

Lemma 3.1. We have

Proof. For every $u\in S_{a}$, by [9, Lemma 5.3], there exists a unique $t_{p, u}\in\mathbb{R}$ such that $t_{p, u}\star u\in\mathcal{P}_{a, p}^-$, that is

and

Since $q\gamma_{q} < 2$ and $p\gamma_{p} > 2$, by (3.2), we have

We know

the Lebesgue dominated convergence theorem implies $\lim_{p\rightarrow 2^*}\lVert u\rVert_{p}^p = \lVert u\rVert_{2^*}^{2^*}$. Therefore, there exists two constants $t_{2} > t_{1}$ independent of $p$ such that $t_{p, u}\in [t_{1}, t_{2}]$ when $p$ close enough to $2^*$. Up to a subsequence, we assume that $t_{p, u}\rightarrow t_{u}$ as $p\rightarrow 2^*$.

Let $p\rightarrow 2^*$, by (3.2) and (3.3), we obtain

and

which implies $t_{u}\star u\in\mathcal{P}_{a, 2^*}^-\cup\mathcal{P}_{a, 2^*}^0$. From [10, Page 20], we know $\mathcal{P}_{a, 2^*}^0 = \emptyset$ and hence $t_{u}\star u\in\mathcal{P}_{a, 2^*}^-$.

By the definition of $m^{-}(a, p)$, we have

which implies

By the definition of $m^{-}(a, 2^*)$ and the arbitrary of $u$, we know the conclusion holds.

The proof of the following two lemmas can be found in [11, Lemmas 3.1, 3.2].

Lemma 3.2. $0 < m^{-}(a, 2^*) < m^{+}(a, 2^*) +\frac{1}{N}S^{\frac{N}{2}}$.

Lemma 3.3. $m^{\pm}(a, 2^*)$ is non-increasing for $0 < a < (\mu^{-1}\alpha(N, q))^{\frac{1}{q(1-\gamma_{q})}}$.

Let $2+\frac{4}{N} < p_{n} < 2^*$ and $p_{n}\rightarrow 2^*$ as $n\rightarrow \infty$. By [9, Theorem 1.3 (ii)], there exists mountain-pass type solutions $\{u_{n}\}\in\mathcal{P}_{a, p_{n}}^-$ for $E_{p_{n}}|_{S_{a}}$ which are positive, radially symmetric such that $E_{p_{n}}(u_{n}) = m^{-}(a, p_{n})$.

Lemma 3.4. $\{u_{n}\}$ is bounded in $H^1(\mathbb{R}^N)$.

Proof. By Lemma 3.1, we have

for $n$ sufficiently large. Since $q\gamma_{q} < 2$, we know $\{u_{n}\}$ is bounded in $H^1(\mathbb{R}^N)$.

Up to a subsequence, there exists $u\in H^1(\mathbb{R}^N)$ such that $u_{n}\rightharpoonup u$ in $H^1(\mathbb{R}^N)$, $u_{n}\rightarrow u$ in $L^r(\mathbb{R}^N)$ with $r\in (2, 2^*)$ and $u_{n}\rightarrow u$ a.e. in $\mathbb{R}^N$. Our main goal is to prove that $u$ is the mountain-pass type solution of $E_{2^*}|_{S_{a}}$. Next, we prove $u$ satisfies (1.2).

Lemma 3.5. There exists $\lambda\leqslant 0$ such that

and $\lambda = 0$ if and only if $u\equiv 0$.

Proof. By [9, Theorem 1.3], there exists $\lambda_{n} < 0$ such that

which together with $u_{n}\in\mathcal{P}_{a, p_{n}}^-$, implies

Let $n\rightarrow \infty$, by (3.6), we have that there exists a $\lambda\leqslant 0$ such that $\lambda_{n}\rightarrow \lambda$ and

Therefore, $\lambda = 0$ if and only if $u\equiv 0$.

For every $\psi\in H^1(\mathbb{R}^N)$, since $\{u_{n}^{2^*-1}\}$ is bounded in $L^{\frac{2^*}{2^*-1}}(\mathbb{R}^N)$ and $\{u_{n}^{q-1}\}$ is bounded in $L^{\frac{q}{q-1}}(\mathbb{R}^N)$, by weak convergence, we have

as $n\rightarrow \infty$. We know that

Therefore, the Lebesgue dominated convergence theorem implies

By (3.5), we have

as $n\rightarrow \infty$, which implies $u$ satisfies (3.4).

Set $\lVert u\rVert_{2} = c\leqslant a$. By Pohozaev identity, we know $u\in\mathcal{P}_{c, 2^*}$. Thus, by [10, (4.2)],

Lemma 3.6. We have $u_{n}\rightarrow u$ in $D^{1, 2}(\mathbb{R}^N)$ as $n\rightarrow \infty$.

Proof. Let $v_{n} = u_{n}-u\rightharpoonup 0$ in $H^1(\mathbb{R}^N)$ as $n\rightarrow \infty$. The Brézis-Lieb Lemma^[22] implies

and

Since $u_{n}\in\mathcal{P}_{a, p_{n}}^{-}$, by the Young inequality, we know

Therefore,

We assume that $\lVert\nabla v_{n}\rVert_{2}^2\rightarrow l$ as $n\rightarrow \infty$. By (3.7), we know $l = 0$ or $l\geqslant S^{\frac{N}{2}}$. If $l\geqslant S^{\frac{N}{2}}$, by Lemmas 3.1 and 3.3, we have

which contradicts with Lemma 3.2. Thus, we obtain $l = 0$ which implies $u_{n}\rightarrow u$ in $D^{1, 2}(\mathbb{R}^N)$.

Lemma 3.7. We have $u\not\equiv 0$.

Proof. Since $u_{n}\in\mathcal{P}_{a, p_{n}}^-$, we have

and

Combining (3.8) and (3.9), there is

That is

Let $n\rightarrow \infty$, by Lemma 2.2, we obtain

which implies $u\not\equiv 0$.

Remark 3.1. By Lemma 3.5, we have $\lambda<0$. $

Lemma 3.8. $u_{n}\rightarrow u$ in $L^2(\mathbb{R}^N)$ as $n\rightarrow \infty$ and hence $u\in S_{a}$.

Proof. The idea of this proof comes from the proof of [10, Proposition 3.1]. Multiplying $u_{n}-u$ on both sides of (3.4) and (3.5), integrating and then subtract, we obtain

By Lemma 3.6, since $u_{n}\rightarrow u$ in $D^{1, 2}(\mathbb{R}^N)$, the first, third and fourth integrals of (3.10) tend to $0$ as $n\rightarrow \infty$. Therefore,

which implies $u_{n}\rightarrow u$ in $L^2(\mathbb{R}^N)$.

Remark 3.2 From Lemma 8, we get that $u_{n}\rightarrow u$ in $H^1(\mathbb{R}^N)$ as $n\rightarrow \infty$.

Proof of Theorem 1.1. By Lemma 3.5 and Remark 3.2, we just need to prove that $E_{2^*}(u) = m^{-}(a,2^*)$ and $u\in\mathcal{P}_{a,2^*}^-$. Since $u_{n}\rightarrow u$ in $D^{1,2}(\mathbb{R}^N)$, by the Sobolev inequality, $u_{n}\rightarrow u$ in $L^{2^*}(\mathbb{R}^N)$. Therefore, combining Lemma 3.1, we have

which implies $E_{2^*}(u) = m^-(a, 2^*)$. Let $n\rightarrow \infty$, by (3.8) and (3.9), we know that $u\in\mathcal{P}_{a, 2^*}^-\cup\mathcal{P}_{a, 2^*}^0$. Since $\mathcal{P}_{a, 2^*}^{0} = \emptyset$ (see [10, Page 20]), there is $u\in\mathcal{P}_{a, 2^*}^-$.

Remark 3.3. From (3.11), we can get that $\lim_{p\rightarrow 2^*}m^{-}(a, p) = m^{-}(a, 2^*)$.

Acknowledgments

The second author was supported by Postgraduate Research an Innovation Project of Chongqing (No. CYS23184).

Use of AI tools declaration

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

Conflict of interest

The authors declare there is no conflict of interest.