Abstract
In this paper, we study the relation between the least energy levels and between the minimizers of the following minimization problems
and
We show that as \(\sigma \rightarrow 0^+\), the minimizers for \(E_\sigma (\rho )\), after rescaling, converge to the minimizers of \(Z(\rho )\). Besides, we also give estimates for \(E_\sigma (\rho )\) and the corresponding Lagrange multiplier when \(\sigma \) is small.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The aim of this work is to explore the asymptotic behaviors of the minimizer and the least energy level to the following minimization problem with \(L^2\)-constraint:
where \(N\ge 1\), \(\rho >0\), and the power \(\sigma \) satisfies the so-called \(L^2\)-subcritical condition that \(0<\sigma <\frac{2}{N}\) (see [5]). By the classical results from [1, 2, 5, 8, 10], for each \(\rho >0\), we know that \(E_{\sigma }(\rho )\) is achieved at some \(w_{\sigma }(x)=w_{\sigma }(|x|)>0\) depending on \(\sigma \) and \(\rho \). Moreover, the minimizer \(w_{\sigma }(x)\) is unique up to translations, decreases in \(r=|x|\), and decays exponentially at infinity. The Euler-Lagrange equation corresponding to problem (1.1) is as follows:
where \(\mu =\mu _\sigma (\rho )\in \mathbb {R}\) appears as a Lagrange multiplier depending on \(\sigma \) and \(\rho \). Solutions with prescribed \(L^2\)-norms are known as normalized solutions. \(w_{\sigma } \) is a normalized ground state solution of (1.2), since it is a nontrivial solution to (1.2) having the least energy \(E_{\sigma }(\rho )\). Problem (1.1) and (1.2) are motivated in particular by the search for stationary states in nonlinear Schrödinger equation, that is, the following time-dependent nonlinear Schrödinger equation
which appears in nonlinear optics and the theory of Bose-Einstein condensates (see [7, 9, 15]). The constraint \(\int _{\mathbb {R}^N}w^2\textrm{d}x=\rho \) in the stationary problem is introduced due to the mass conservation property of the time-dependent nonlinear Schrödinger equation. In applications, the prescribed mass represents the power supply in nonlinear optics or the number of particles in Bose-Einstein condensates. The problem (1.2) also appears in the Mean Field Games theory in Lasry and Lions [11] as a case of the mean field limit equations in a stationary setting.
The present paper is invoked by [19] that uncovers a relation between power-law nonlinear scalar field equations and logarithmic-law scalar field equations. In [19], Wang and Zhang consider the following power-law nonlinear Schrödinger equation:
and logarithmic-law nonlinear Schrödinger equation:
They show that as \(p\downarrow 2\), the ground state solutions of (1.3), after a unique rescaling, converge to the ground state solutions of (1.4). The logarithmic nonlinear Schrödinger equation was introduced as an important model in quantum physics. It admits plenty of applications related to quantum mechanics, quantum optics, nuclear physics, transport and diffusion phenomena, theory of superfluidity and Bose–Einstein condensation, see [3, 4, 16, 17, 19] and the references therein. d’Avenia-Montefusco-Squassina[6] and Troy[16] have proved that the ground state solution of (1.4) is unique up to translations and is given by
Therefore, it is easy to see that the \(L^2\)-constrained minimization problem
is achieved at
The logarithmic nonlinear Schrödinger equation corresponding to problem (1.5) is as follows:
here \(\lambda \in \mathbb {R}\) appears as a Lagrange multiplier dependent of \(\rho \). It is obvious that \(v^0(x)\) is a normalized ground state solution of (1.6). A direct calculation shows that the Lagrange multiplier is
and the least energy to (1.6) is
These explicit formulas give clear information about the Lagrange multiplier and the ground state energy to (1.6). In contrast, very little is known about \(\mu _\sigma \) and \(E_\sigma \) of (1.2) except for some obvious knowledge that, as \(\sigma \rightarrow 0\), \(E_\sigma (\rho )\rightarrow -\rho /2\) and \(\mu _\sigma \rightarrow 1\). Further information is rather difficult to obtain.
In order to establish the relation between the two \(L^2\)-constrained minimization problems (1.1) and (1.5), and give estimates on \(E_\sigma (\rho )\) and \(\mu _\sigma (\rho )\), we are dedicated to proving the convergence of \(w_{\sigma }(x)\) to \(v^0(x)\) as \(\sigma \rightarrow 0\).
Theorem 1.1
Let \(v_{\sigma }(x)=\sigma ^{-\frac{N}{2(2-\sigma N)}}w_{\sigma }(\sigma ^{-\frac{1}{2-\sigma N}}x)\). Then, as \(\sigma \rightarrow 0\), it holds that \(v_{\sigma }\rightarrow v^0\) strongly in \(H^1(\mathbb {R}^N)\) and in \(C^{2,\alpha }(\mathbb {R}^N)\) for any \(\alpha \in (0,1)\). Moreover, we have
In comparison to the recent and interesting work [19] by Wang and Zhang, the major novelty here is given by the mass constraint, that was not considered in [19]. In [19], the authors study the asymptotic behaviors of the least energy level and ground state solutions of probelm (1.3) with a prescribed \(\lambda \). After a scaling \(u_p(x)=\lambda ^{-\frac{1}{p-2}}v_p(\frac{x}{\sqrt{p-2}})\)(see [19], Theorem 1.1), as \(p\downarrow 2\), they derive the convergence of \(u_p(x)\) to U. However in this paper, \(\mu \in \mathbb {R}\) appears as a Lagrange multiplier, which varies as \(\sigma \rightarrow 0^+\). So it is difficult to use the result in [19] to obtain the asymptotic behaviors of \((\mu _\sigma , w_\sigma )\) solving Eq.(1.2). We introduce a new scaling in which \(\mu \) does not appear to achieve these behaviors.
Remark 1.1
We remark that \(\rho ^{-\frac{2\sigma }{2-\sigma N}}\mu _{\sigma }(\rho )\), \(\rho ^{-1-\frac{2\sigma }{2-\sigma N}}E_\sigma (\rho )\) are quantities independent of \(\rho \), see (2.4) and (2.7). Moreover, when \(\rho \) belongs to a compact subset of \((0,+\infty )\), there hold
The paper is organized as follows: In Sect. 2, we establish the work space and give some preliminaries which will be used in the proof of the main theorems; Sect. 3 is devoted to the proof of Theorem 1.1.
2 Preliminaries
Throughout this paper, we use the following notations:
-
\(H^1(\mathbb {R}^N)\) is the usual Sobolev space with the following inner product and norm
$$\begin{aligned} (u,v):=\int _{\mathbb {R}^N}\nabla u\nabla v+uv\textrm{d}x,\quad \left\| u\right\| : =\sqrt{(u,u)},\quad \forall u,v\in H^1(\mathbb {R}^N). \end{aligned}$$\(H_{rad}^1(\mathbb {R}^N)\) denotes the space \(\left\{ u\in H^1(\mathbb {R}^N)|u(x)=u(|x|)\right\} .\)
-
\(L^p(\mathbb {R}^N)\) (\(1\le p<\infty \)) is the Lebesgue space with the norm \(\left| u\right| _p =\big (\int _{\mathbb {R}^N}|u|^p\textrm{d}x\big )^{\frac{1}{p}}\).
-
\(o_\sigma (1)\) denotes an infinitesimal with \(o_\sigma (1)\rightarrow 0\) as \(\sigma \rightarrow 0\).
-
\(C(a_1,a_2,\ldots ,a_n)\) denotes any positive constant that depends on \(a_1,a_2,\ldots ,a_n\).
-
For \(R>0\), \(B_{R}(0)\) denotes the ball of radius R centered at 0. \(B_{R}^c(0)\) denotes the set \(\mathbb {R}^N\setminus B_{R}(0)\).
Consider \(I_\sigma :H^1(\mathbb {R}^N)\rightarrow \mathbb {R}\) defined by
We will perform some scaling on w so that we can approach the limit functional \(J: H^1(\mathbb {R}^N)\rightarrow \mathbb {R}\cup \{+\infty \}\) given by
First, following [19], for \(w\in H^1(\mathbb {R}^N)\) and \(|w|_2^2=\rho \), set \(u(x)=w(\frac{1}{\sqrt{\sigma }}x)\). Then we get \(|u|_2^2=\rho \sigma ^{\frac{N}{2}}\) and
Next, we use a re-scaling that shifts the \(L^2\) norm of functions and preserves some homogeneity on the functional (see also [18, 20]). For each \(s>0\), letting \(u(x)=s^{\frac{1}{2-\sigma N}}v(s^{\frac{\sigma }{2-\sigma N}}x)\), we have
Setting \(s=\sigma ^{\frac{N}{2}}\), we have \(|v|_2^2=\rho \). By (2.1), we arrive at
where
Note that in (2.2)
Remark 2.1
If we set \(s=\rho \sigma ^{\frac{N}{2}}\), then \(|v|_2^2=1\) and
Setting , we introduce another \(L^2\)-constrained minimization problem
Then, by (2.2) and Remark 2.1, it is clear that
Since for any \(\varphi \in C_0^{\infty }(\mathbb {R}^N)\),
the minimizer corresponding to problem (2.3) satisfies:
Here \(\lambda _\sigma =\lambda _\sigma (\rho )\in \mathbb {R}\) appears as a Lagrange multiplier. From [20, Lemma 2.1], we know \((\lambda _\sigma , v_\sigma )\) is unique up to translations. So \(\lambda _\sigma \) depends only on N, \(\sigma \) and \(\rho \). It is easy to verify that
In the sequel, we will compare the above equation (2.6) with the logarithmic Schrödinger equation (1.6).
Note that \(I_\sigma \) and \(J_\sigma \) are \(C^2\) functionals but J is not continuous(see [6] or [17]). In fact, by using the following standard logarithmic Sobolev inequality(see Lieb and Loss [12])
where the equality holds if and only if \(u(x)=e^{-\frac{\pi |x|^2}{2c^2}}\), it is obvious that \(\int _{\mathbb {R}^N}u^2\log u^2\textrm{d}x < +\infty \) for all \(u\in H^1(\mathbb {R}^N)\). Indeed there exists \(u\in H^1(\mathbb {R}^N)\) such that \(\int _{\mathbb {R}^N}u^2\log u^2\textrm{d}x=-\infty \) (see [13]). Thus, in general, the functional J fails to be finite and lacks \(C^1\)-smoothness on \(H^1(\mathbb {R}^N)\). However as in [17], the \(L^2\)-constrained minimization problem (1.5) can also be considered in the following space
In addition, we give some important notations:
Definition 2.1
-
(1)
We say \(C_0^{\infty }(\mathbb {R}^N)\) is dense in \(\mathcal {D}\) in the following sense: for any \( v\in \mathcal D\), there exists a sequence \(\varphi _n\in C_0^\infty (\mathbb {R}^N)\), such that
$$\begin{aligned} \Vert \varphi _n - v\Vert \rightarrow 0\quad \text{ and }\quad \int _{\mathbb {R}^N} \varphi _n^2\log \varphi _n^2\textrm{d}x \rightarrow \int _{\mathbb {R}^N} v^2\log v^2\textrm{d}x. \end{aligned}$$ -
(2)
For \(v, \varphi \in \mathcal D\), we define
$$\begin{aligned} J'(v)\varphi :=\int _{\mathbb {R}^N}\nabla v\nabla \varphi \textrm{d}x-\int _{\mathbb {R}^N}v\varphi (1+\log v^2)\textrm{d}x \end{aligned}$$ -
(3)
We say \(v\in \mathcal {D}\cap \mathcal M_\rho \) is a critical point of J on \(\mathcal D\cap \mathcal {M}_\rho \) if and only if \( J'(v)\varphi =0\) for every \(\varphi \in C^{\infty }_0(\mathbb {R}^N)\) such that \(\int _{\mathbb {R}^N} v\varphi =0\).
Remark 2.2
Note that \(v\in \mathcal {D}\cap \mathcal {M}_\rho \) is a critical point of J on \(\mathcal D\cap \mathcal {M}_\rho \) if and only if there is \(\lambda \in \mathbb {R}\) such that
Moreover, \(\lambda =\rho ^{-1}J'(v)v\).
The following lemma (see [19, Lemma 2.1]) describes the behavior of the nonlinear term as \(\sigma \) near 0.
Lemma 2.1
-
(i)
For any \(\sigma '>0\), there exists \(C(\sigma ')>0\) such that
$$\begin{aligned} \frac{x^{2\sigma }-1}{\sigma }\le C(\sigma ')x^{2\sigma '} \end{aligned}$$holds for all \(\sigma \in (0,\sigma ')\) and \(x\ge 0 \).
-
(ii)
Let \(s>0\), \(\sigma >0\), then
$$\begin{aligned} \frac{x^s(x^\sigma -1)}{\sigma }\rightarrow x^s\log x \quad \text {in}\quad C_{loc}^{m,\alpha }[0,+\infty ) \end{aligned}$$as \(\sigma \rightarrow 0\), where m is the largest integer with \(m<s\), and \(\alpha \in (0,s-m)\).
3 Proofs of the main results
We first consider the case \(\rho =1\) and write \(Z_\sigma =Z_\sigma (1)\).
Lemma 3.1
\(Z_\sigma \) is achieved at
Moreover, as \(\sigma \rightarrow 0^+\), \(Z_\sigma \), \(\Vert v_{\sigma }\Vert \), and \(\int _{\mathbb {R}^N}\sigma ^{-1}|| v_{\sigma }|^{2\sigma +2}- v_{\sigma }^2| \) are bounded.
Proof
Recalling (2.2), for \(w\in H^1(\mathbb {R}^N)\) and \(|w|_2^2=1\), by rescaling
we derive that \(|v|_2^2=1\) and
Then,
implying that
and \(E_{\sigma }(1)\) is attained at \(w_{\sigma }\) if and only if \(Z_\sigma (1)\) is attained at \(v_\sigma \).
We next prove that \(Z_{\sigma } \) is bounded as \(\sigma \rightarrow 0^+\). Fixing any \(v\in C_0^{\infty }(\mathbb {R}^N)\setminus \{0\}\) satisfying \(\int _{\mathbb {R}^N}|{v}|^{2}\textrm{d}x=1\), from Lemma 2.1(ii), one gets \(J_\sigma (v) \rightarrow J(v)\) as \(\sigma \rightarrow 0\). Therefore, \(Z_{\sigma }(1)\) is bounded from above.
Applying Lemma 2.1(i) and Gagliardo-Nirenberg inequality
where \(\sigma '\) is a fixed constant such that \(0<\sigma 'N<2\), we obtain
Then (3.1) implies that \(Z_{\sigma }\) is bounded from below. Thus, \(Z_{\sigma }\) is bounded. In addition, the boundedness of \(Z_{\sigma }\) and (3.1) assert that \(|\nabla v_{\sigma }|_2\) is bounded. Together with \(|v_{\sigma }|_2^2=1\), we derive that \(\Vert v_{\sigma }\Vert \) is bounded. By the Sobolev embedding \(H^1(\mathbb {R}^N)\hookrightarrow L^{p}(\mathbb {R}^N)\) for \(p\in (2,2^*)\), where \(2^*=\frac{2N}{N-2}\) for \(N\ge 3\) and \(2^*=+\infty \) for \(N=1,2\), we have
By
we know \(\int _{\mathbb {R}^N}\sigma ^{-1}(| v_{\sigma }|^{2\sigma +2}- v_{\sigma }^2)_- \) is bounded. Thus, \( \int _{\mathbb {R}^N}\sigma ^{-1}|| v_{\sigma }|^{2\sigma +2}- v_{\sigma }^2| \) is bounded. Then we complete the proof. \(\square \)
Theorem 3.1
As \(\sigma \rightarrow 0^+\), we have \(\lambda _{\sigma }\rightarrow \lambda ^0\), \(Z_\sigma (1)\rightarrow Z(1)\), and \(v_{\sigma }\rightarrow v^0\) strongly in \(H^1(\mathbb {R}^N).\)
Proof
Since \(v_{\sigma }\) satisfies (2.6) with \(\rho =1\), i.e.
by Lemma 3.1, we know that
is bounded.
We next re-scale \(v_{\sigma }\) in order to avoid the possible difficulties that \(\lambda _{\sigma }<0\) poses to the subsequent proofs. Setting \(\tilde{v}_{\sigma }=av_{\sigma }\), from (3.2), we can get
where
Since
we can fix \(a>0\) sufficiently large such that \(\mu _{\sigma }>1\) for every small \(\sigma \). By Lemma 3.1 and (3.3) and (3.5), \(\Vert \tilde{v}_{\sigma }\Vert \) and \(\mu _{\sigma }\) are both bounded. Then up to a subsequence, we assume
and \(\mu _{\sigma }\rightarrow \mu ^0\in [1,+\infty )\). If \(N\ge 2\), then by the radial lemma of Strauss [14], \(\tilde{v}_{\sigma }(x)<1\), \(|x|\ge R\) for some R independent of \(\sigma \). If \(N=1\), we have \(\tilde{v}_{\sigma }\rightarrow \tilde{v}\) in \(C_{loc}(\mathbb {R})\). Then we also get \(\tilde{v}_{\sigma }(x)<1\) when \(|x|\ge R\) for some R independent of \(\sigma \). Hence for \(N\ge 1\), \(\tilde{v}_{\sigma }\) satisfies \(-\Delta \tilde{v}_{\sigma }+\tilde{v}_{\sigma }\le 0\) in \(\mathbb {R}^N\setminus B_R(0)\). By comparison theorem, we obtain
for \(C, c>0\) independent of \(\sigma \). Necessarily,
Especially,
Multiplying (3.4) by \(\tilde{v_\sigma }-\tilde{v}\) and integrating, we get
Taking limits as \(\sigma \rightarrow 0^+\), we get
It follows that \(\lim _{\sigma \rightarrow 0^+}\Vert \tilde{v_\sigma }\Vert =\Vert \tilde{v}\Vert \). Combining with \(\tilde{v_\sigma }\rightharpoonup \tilde{v}\) in \(H^1(\mathbb {R}^N)\), we deduce that \(\tilde{v_\sigma }\rightarrow \tilde{v}\) strongly in \(H^1(\mathbb {R}^N)\). Up to a translation, we can assume \(\tilde{v}(0)=\max _{\mathbb {R}^N}\tilde{v}\). Since \((\mu ^0, \tilde{v})\) solves
we conclude that \((\mu ^0, \tilde{v})\) is unique. Therefore, the convergence \((\mu _\sigma , \tilde{v_\sigma })\rightarrow (\mu ^0, \tilde{v})\) is independent of subsequences.
By uniqueness of the solution to (2.6), we derive that \(\mu ^0=\lambda ^0+2\log a\) and \(\tilde{v}=av^0\). Thus, \(\lambda _\sigma \rightarrow \lambda ^0\), and \(v_\sigma \rightarrow v^0\) strongly in \(H^1(\mathbb {R}^N)\). \(\square \)
Proof of Theorem 1.1
Combining (2.2) with Lemma 3.1, one can show that
and \(Z_\sigma (\rho )\) is attained at \(v_\sigma (x)=\sigma ^{-\frac{N}{2(2-\sigma N)}}w_\sigma (\sigma ^{-\frac{1}{2-\sigma N}}x)\in \mathcal M_\rho \). By arguments similar to those in the proof of Theorem 3.1, we also have \(\lambda {_\sigma }\rightarrow \lambda ^0\) and \(v_\sigma \rightarrow v^0\) strongly in \(H^1(\mathbb {R}^N)\) as \(\sigma \rightarrow 0\). By regularity theory, it follows that \(v_\sigma \rightarrow v^0\) in \(C^{2,\alpha }(\mathbb {R}^N)\), \(\alpha \in (0,1)\). Applying Lemma 2.1(ii) again, we deduce that \(J_\sigma (v_\sigma )\rightarrow J(v^0)\) as \(\sigma \rightarrow 0\). That is, \(Z_{\sigma }(\rho )\rightarrow Z{(\rho )}\) as \(\sigma \rightarrow 0\). Hence, by (2.7), one gets
and by (2.4),
\(\square \)
References
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, I existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)
Berestycki, H., Lions, P.L.: Nonlinear scalar field equations, II existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82(4), 347–375 (1983)
BiaŁynicki-Birula, I., Mycielski, J.: Wave equations with logarithmic nonlinearities. Bull. Acad. Pol. Sci. Cl 3(23), 461–466 (1975)
BiaŁynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys. 100(12), 62–93 (1976)
Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)
d’Avenia, P., Montefusco, E., Squassina, M.: On the logarithmic Schrödinger equation. Commun. Contemp. Math. 16(2), 1350032 (2014)
Frantzeskakis, D.J.: Dark solitons in atomic Bose-Einstein condensates: from theory to experiments. J. Phys. A 43(21), 213001, 68 (2010)
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)
Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-Gonzalez, R. (eds.): Springer-Verlag, Berlin (2008)
Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u + u^{p} = 0\) in \({\mathbb{R} }^n\). Arch. Ration. Mech. Anal. 105(3), 243–266 (1989)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence (2001)
Shuai, W.: Multiple solutions for logarithmic Schrödinger equations. Nonlinearity 32(6), 2201–2225 (2019)
Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)
Timmermans, E.: Phase separation of Bose-Einstein condensates. Phys. Rev. Lett. 81(26), 5718–5721 (1998)
Troy, W.C.: Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation. Arch. Ration. Mech. Anal. 222, 1581–1600 (2016)
Tanaka, K., Zhang, C.: Multi-bump solutions for logarithmic Schrödinger equations, Calc. Var. Partial Differential Equations 56 (2017)
Tang, Z., Zhang, C., Zhang, L., Zhou, L.: Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well. Nonlinearity 35(8), 4624–4658 (2022)
Wang, Z.-Q., Zhang, C.: Convergence from power-law to logarithm-law in nonlinear scalar field equations. Arch. Ration. Mech. Anal. 231, 45–61 (2019)
Zhang, C., Zhang, X.: Normalized multi-bump solutions of nonlinear Schrödinger equations via variational approach. Calc. Var. Partial Differ. Equ. 61(2), 57, 20 (2022)
Funding
This work was supported by National Natural Science Foundation of China (Grant Number: 12001044).
Author information
Authors and Affiliations
Contributions
LZ wrote the main manuscript text and CZ revised the introduction and theorem 3.1. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, L., Zhang, C. The asymptotic behaviors of normalized ground states for nonlinear Schrödinger equations. Nonlinear Differ. Equ. Appl. 30, 44 (2023). https://doi.org/10.1007/s00030-023-00853-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-023-00853-z