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Ground states for the nonlinear Schrödinger equation under a general trapping potential

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Abstract

The classical Schrödinger equation with a harmonic trap potential \(V(x)=|x|^2\), describing the quantum harmonic oscillator, has been studied quite extensively in the last 20 years. Its ground states are bell-shaped and unique, among localized positive solutions. In addition, they have been shown to be non-degenerate and (strongly) orbitally stable. All of these results, produced over the course of many publications and multiple authors, rely on ODE methods specifically designed for the Laplacian and the power function potential. In this article, we provide a wide generalization of these results. More specifically, we assume sub-Laplacian fractional dispersion and a very general form of the trapping potential V, with the driving linear operator in the form \({{\mathscr {H}}}=(-\Delta )^s+V, 0<s\le 1\). We show that the normalized waves of such semilinear fractional Schrödinger equation exist, and they are bell-shaped, provided that the nonlinearity is of the form \(|u|^{p-1} u, p<1+\frac{4 s}{n}\). In addition, we show that such waves are non-degenerate and strongly orbitally stable. Most of these results are new even in the classical case \({{\mathscr {H}}}=-\Delta +V\), where V is a general trapping potential considered herein.

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Notes

  1. The requirement for at most polynomial growth of V is likely just a technicality, but we prefer to enforce it, due to the difficulties with the space of test functions, should V has faster growth.

  2. In non-dimensionalized variables.

  3. Although it looks as if this result has not been stated explicitly in the literature.

  4. Due to the polynomial growth assumption for V, Schwartz functions are a reliable dense set in all the spaces that we introduce.

  5. For the purposes of the derivation of the Euler–Lagrange equation, the operator \((-\Delta )^s\) applied on \(\phi \) should be understood in a distributional sense, since a priori, we only know that \(\phi \in H^s({{\mathbb {R}}}^n)\). Eventually, we have that \(\phi \in H^{2s}({{\mathbb {R}}}^n)\), so this will not be an issue.

  6. Here, recall that due to Proposition 4, \(\phi \in C^1({{\mathbb {R}}}^n)\), and so \(\phi \in C^1(0, \infty )\) as a function of the radial variable.

  7. Note that here, the a priori information is only \(\psi _1, \Psi _{1,n}\in H^s({{\mathbb {R}}}^n)\), so our functions are not even known to be continuous, unless \(s>\frac{n}{2}\). On the other hand, the property \(\psi \) is positive on an interval \((r_0, \infty )\) is easily tested against a positive test function. That is \(\psi >0\) on an interval I, if for every non-negative \(C^\infty _0(I)\) function, we have \(\langle \psi ,\chi \rangle >0\).

  8. In fact, we can conclude that \(\psi _1\) is both positive and decreasing in \((0, \infty )\).

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Acknowledgements

We would like to thank our frequent collaborator Sevdzhan Hakkaev for numerous insightful conversations on these topics.

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Correspondence to Milena Stanislavova.

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Stanislavova is partially supported by NSF-DMS under Grant # 1516245. Stefanov is partially supported by NSF-DMS under Grant # 1908626.

Appendix A: A posteriori smoothness and decay: Proof of Proposition 4

Appendix A: A posteriori smoothness and decay: Proof of Proposition 4

We start with the a priori information from Proposition 3, that is \(\phi \) is bell-shaped and in the class \(\phi \in H^s\cap L^2(V(x) \mathrm{d}x)\), together with the fact that \(\phi \) is a weak solution of (3.1).

In order to obtain bootstrap this information, we need a representation of \(\phi \) from the Euler–Lagrange PDE. Unfortunately, \(\phi \) is still only a weak solution of (3.1), as we have pointed out. Instead, define for large enough N,

$$\begin{aligned} {\tilde{\phi }}:= ( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ^p+N\phi ]. \end{aligned}$$

Heuristically, this is the solution of the (3.1), if \(\phi \) were a solution in a stronger sense. In fact, it is not even immediately clear in what sense is \({\tilde{\phi }}\) even defined. Clearly, while

$$\begin{aligned} \Vert ( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ]\Vert _{L^2}\le C\Vert \phi \Vert _{L^2} \end{aligned}$$

is under control, it is not as easy to control \(( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ^p]\), since the a priori information on \(\phi ^p\) is very weak. Instead, for \(n\le 4s\), we can bound by (2.5) and Sobolev embedding

$$\begin{aligned} \Vert ( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ^p]\Vert _{L^2({{\mathbb {R}}}^n)}\le C \Vert \phi ^p\Vert _{H^{-2s}}\le C \Vert \phi ^p\Vert _{L^{\frac{p+1}{p}}} = C \Vert \phi \Vert _{L^{p+1}}^p. \end{aligned}$$

while for \(n>4s\), we bound by (2.5) and by repeated application of Sobolev embedding

$$\begin{aligned} \Vert ( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ^p]\Vert _{L^2({{\mathbb {R}}}^n)}\le C \Vert \phi ^p\Vert _{H^{-2s}}\le C \Vert \phi ^p\Vert _{L^{2}} \le C \Vert \phi \Vert _{H^s({{\mathbb {R}}}^n)}^p. \end{aligned}$$

So, \({\tilde{\phi }}\) is well-defined as an \(L^2({{\mathbb {R}}}^n)\) function. Consider a test function \(h\in H^{2s}\cap L^2(V^2(x) \mathrm{d}x)\),

$$\begin{aligned} \langle {\tilde{\phi }},( (-\Delta )^s+V +\omega _\lambda +N) h \rangle = \langle \phi ^p+N\phi ,h \rangle =\langle \phi ,(-\Delta )^s+V +\omega _\lambda +N) h \rangle . \end{aligned}$$

It follows that \(\langle \phi -{\tilde{\phi }},(-\Delta )^s+V +\omega _\lambda +N) h \rangle =0\). Since the set \(\{ (-\Delta )^s+V +\omega _\lambda +N) h: h\in H^{2s}\cap L^2(V^2(x) \mathrm{d}x)\}\) is dense in \(L^2\), we have that \(\phi ={\tilde{\phi }}\) or

$$\begin{aligned} \phi = ( (-\Delta )^s+V +\omega _\lambda +N)^{-1}[\phi ^p+N\phi ]. \end{aligned}$$
(A.1)

We now run a bootstrapping procedure, which will ultimately establish that \(\phi \in H^{2s}({{\mathbf {R}}}^d)\). Starting with \(\alpha _0=s\), we define \(\alpha _{k+1}\), as long as \(\alpha _k<2s\). We have for \(\alpha : \alpha _k<\alpha \le 2s\), by Sobolev embedding, (2.7) and Kato-Ponce estimates

$$\begin{aligned} \Vert \phi \Vert _{H^{\alpha }}\le & {} C[\Vert \phi \Vert _{L^2}+\Vert \phi ^p\Vert _{H^{\alpha -2s}}]\le C[\Vert \phi \Vert _{L^2}+\Vert \phi \Vert _{H^{\alpha _k}} \Vert \phi ^{p-1}\Vert _{L^{\frac{n}{2s+\alpha _k-\alpha }}}]\\= & {} C[\Vert \phi \Vert _{L^2}+\Vert \phi \Vert _{H^{\alpha _k}} \Vert \phi \Vert _{L^{\frac{n(p-1)}{2s+\alpha _k-\alpha }}}^{p-1}]. \end{aligned}$$

In the last term, if we make sure that \(\frac{n(p-1)}{2s+\alpha _k-\alpha }\le p+1\), we will have control of the right-hand side. Given the restriction \(p<1+\frac{4s}{n}\), this would be satisfied, if

$$\begin{aligned} \alpha -\alpha _k\le \frac{4s^2}{n+2s}. \end{aligned}$$

So, we define \(\alpha _{k+1}:=\min (2s, \alpha _k+\frac{4s^2}{n+2s})\), whence we conclude that \(\phi \in H^{\alpha _{k}}\) for each k. Clearly, in finitely many iterations, we will reach \(\phi \in H^{2s}({{\mathbf {R}}}^d)\).

Furthermore, \(\phi ^p\in L^2\), since

$$\begin{aligned} \Vert \phi ^p\Vert _{L^2}=\Vert \phi \Vert _{L^{2p}}^p\le C \Vert \phi \Vert _{H^{2s}}^p, \end{aligned}$$

since \(p<1+\frac{4s}{n}\). It follows from (2.7) that \(\phi \in H^{2s}\cap L^2(V^2(x) \mathrm{d}x)\) since

$$\begin{aligned} \Vert \phi \Vert _{H^{2s}\cap L^2(V^2(x) \mathrm{d}x)}\le C [\Vert \phi \Vert _{L^2}+ \Vert \phi ^p\Vert _{L^2}]<\infty . \end{aligned}$$

Once we have that \(V \phi \in L^2\), it is easy to bootstrap even further. Indeed, we will have that the expression \(( (-\Delta )^s+\omega +N)^{-1}[(V+N) \phi ]\) makes sense as \(L^2\) function, which is positive everywhere, for N large enough, as convolution of \(G_{\omega +N}>0\) and \((V+N) \phi >0\). Hence, we have

$$\begin{aligned} 0<\phi =( (-\Delta )^s+\omega +N)^{-1}[\phi ^p+2 N \phi - (V+N)\phi ]\le ( (-\Delta )^s+\omega +N)^{-1}[\phi ^p+2 N \phi ] \end{aligned}$$

This last inequality can be now iterated to \(\phi \in L^\infty ({{\mathbb {R}}}^n)\), see p. 1723, [6].

We now aim at extending this further to Lipschitz continuity. To this end, introduce a smooth and even cutoff function \(\chi : supp\chi \subset (-2,2)\), so that \(\chi (x)=1, |x|<1\). Let \(N>>1\) and \(\chi _N(x):=\chi (x/N)\). Multiplying equation (1.2) by the cutoff \(\chi _N\) and \(\phi _N:=\phi (x) \chi _N\), we can rewrite it in the form

$$\begin{aligned} ((-\Delta )^s +\omega +M)\phi _N= - V \phi _N+\phi ^{p}\chi _N+M \phi _N+[(-\Delta )^s, \chi _N]\phi . \end{aligned}$$
(A.2)

for any M. The operator on the left-hand side is invertible for large enough M, and we can write

$$\begin{aligned} \phi _N=((-\Delta )^s +\omega +M)^{-1}[- V \phi _N+\phi ^{p}\chi _N+M \phi _N+[(-\Delta )^s,\chi _N]\phi ]. \end{aligned}$$
(A.3)

According to the Mikhlin multplier’s theorem, \(((-\Delta )^s +\omega +M)^{-1}\) smooths out by 2s derivatives in any Sobolev space \(W^{\alpha , p}, 1<p<\infty \). It follows that for any \(\alpha <2s\),

$$\begin{aligned} \Vert \phi _N\Vert _{W^{\alpha , p}}\le C_{\alpha , p}[ \Vert V \phi _N\Vert _{L^p} +\Vert \phi ^{p}\chi _N\Vert _{L^p} +M \Vert \phi _N\Vert _{L^p}+\Vert [(-\Delta )^s,\chi _N]\phi \Vert _{L^p}\le C_{\alpha , p}, \end{aligned}$$

due to the a priori bounds on \(\Vert \phi \Vert _{L^p}\), and the fact that V is bounded on the support of \(\chi _N\). Note that we also have used a corollary of the commutator estimates to derive \(\Vert [(-\Delta )^s,\chi _N] \phi \Vert _{L^p}\le C_{N,p, {\tilde{p}}} \Vert \phi \Vert _{L^{{\tilde{p}}}}, {\tilde{p}}>p\). It follows that \(\phi _N\in W^{2s, p}, p<\infty \) for each N. If \(2s>1\), there is nothing to do, as \(\phi _N\in W^{1+, p}, p<\infty \), which by Sobolev embedding will imply that \(\phi \in C^1\) as required.

Otherwise, apply \((-\Delta )^s\) to (A.2) and then use the inversion formulas as in (A.3). Since \(\phi _N\in W^{2s, p}\), we see that (recall that \(V\in C^1({{\mathbb {R}}}^n)\))

$$\begin{aligned} (-\Delta )^s [- V \phi _N+\phi ^{p}\chi _N+M \phi _N+[(-\Delta )^s,\chi _N]\phi ] \in L^p, \end{aligned}$$

whence \(\phi _N\in W^{4s, p}\) and so on. This can be bootstrapped, in finitely many steps to the desired outcome \(\phi _N\in W^{1+, p}, p<\infty \), so \(\phi \in C^1\). We omit further details.

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Stanislavova, M., Stefanov, A.G. Ground states for the nonlinear Schrödinger equation under a general trapping potential. J. Evol. Equ. 21, 671–697 (2021). https://doi.org/10.1007/s00028-020-00596-9

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