On the inhomogeneous NLS with inverse-square potential

In this paper, we study the initial value problem (IVP) associated to the inhomogeneous nonlinear Schrödinger equation (INLS_a for short)

where $N\geq 3$, $\mathcal{L}_{a}=-\Delta+\frac{a}{|x|^{2}}$ with $a>-\frac{(N-2)^{2}}{4}$ and $\alpha,b>0$. The equation is called “focusing INLS_a” when $\lambda=+1$ and “defocusing INLS_a” when $\lambda=-1$. The restriction on $a$ comes from the sharp Hardy inequality, namely

which guarantees that $\mathcal{L}_{a}$ is a positive operator. It is a model from various physical contexts, for example, in nonlinear optical systems with spatially dependent interactions (see e.g. [1] and the references therein). In particular, when $a=0$, it can be thought of as modeling inhomogeneities in the medium in which the wave propagates (see for instance [18]). When $b=0$, the equation (1.1) appears in several physical settings, such a quantum field equations or black hole solutions of the Einstein’s equations (see e.g. [17]).

The INLS_a model can be seen as an extension to the Schrödinger equation. For instance, when $a=b=0$ we have the classical nonlinear Schrödinger (denoted by NLS) equation, extensively studied over the three decades. (see [2], [6], [20], [26] and the references therein). The case $b=0$ and $a\neq 0$ is known as the NLS equation with inverse square potential, denoted by NLS_a equation, that is,

This equation has also been intensively studied in recent years (see for instance, [3], [21], [22], [23], [27]). Moreover, when $a=0$ and $b\neq 0$ we have the inhomogeneous NLS equation, denoted by (INLS), i.e.,

which also has received substantial attention recently (see e.g. [16], [11], [12], [4], [8], [21]).

Similarly to the INLS, the INLS_a equation is invariant under the scaling, $u_{\mu}(t,x)=\mu^{\frac{2-b}{\alpha}}u(\mu^{2}t,\mu x)$ for $\mu>0$. A straightforward computation yields

implying that the scale-invariant Sobolev space is $\dot{H}^{s_{c}}(\mathbb{R}^{N})$, with $s_{c}=\frac{N}{2}-\frac{2-b}{\alpha}$, the so called critical Sobolev index. If $s_{c}=0$ (or $\alpha=\frac{4-2b}{N}$) the (IVP) is known as mass-critical or $L^{2}$-critical; if $s_{c}<0$ (or $0<\alpha<\frac{4-2b}{N}$) it is called mass-subcritical or $L^{2}$-subcritical; if $0<s_{c}<2$, (1.1) is known as mass-supercritical and energy-subcritical (or intercritical). In additional, solutions to (1.1) conserve their mass and energy, defined respectively by

When $a=0$, we denote $E_{a}$ by $E_{0}$.

We review now some recent developments in $H^{1}(\mathbb{R}^{N})$, for the particular cases, i.e., NLS_a and INLS models, starting with the NLS_a equation. Okazawa-Suzuki-Yokota [23], by the energy method, showed local well-posedness in the energy-subcritical case, for $N\geq 3$ and $a>-\tfrac{(N-2)^{2}}{4}$. They also proved that the solutions are global if $\lambda=-1$ or $0<\alpha<\frac{4}{N}$ and $\lambda=1$. Zhang-Zheng [27] studied the defocusing case, establishing well posedness and scattering for $\tfrac{4}{N}<\alpha<\tfrac{4}{N-2}$, assuming $a\geq 0$ and $N=3$, or $a\geq-\tfrac{(N-2)^{2}}{4}+\tfrac{4}{(\alpha+2)^{2}}$ and $N\geq 4$. Recently, Killip-Murphy-Visan-Zheng in [21] considered the focusing $3D$ cubic NLS_a. They established well-posedness and scattering in $H^{1}(\mathbb{R}^{N})$ if $a>-\tfrac{1}{4}$. Later, Lu-Miao-Murphy [22] extended the result of [21] to all $L^{2}$-supercritical, energy subcritical nonlinearities, in dimensions, $3\leq N\leq 6$, with

On the other hand, the INLS equation was first studied by Genoud-Stuart [15] via the energy method. For $N\geq 1$ and $0<b<\min\{N,2\}$, they showed local well posedness in $H^{1}(\mathbb{R}^{N})$ for the $H^{1}$-subcritical case and global well possedness in the mass-subcritical case. In the mass-critical case, Genoud in [14] established global well-posedness in $H^{1}(\mathbb{R}^{N})$, provided that the mass of the initial data is below that of the associated ground state. This result was extended in the intercritical case by Farah [10] and he also showed that the solution blows-up in finite time (see also [9]). The second author in [16], using Kato’s method, proved local well-posedness in $H^{1}(\mathbb{R}^{N})$, for the energy subcritical case in dimensions $N\geq 4$ and $0<b<2$. Cho-Lee [7] treated the case $N=3$ for $0<b<\frac{3}{2}$ and Dinh [8] the case $N=2$ for $0<b<1$. Furthermore, in the intercritical case, the second author in [16] also established a small data global theory in $H^{1}(\mathbb{R}^{N})$ for $N\geq 4$, the first author [4] treated the case $N=3$ and Cardoso-Farah-Guzmán [5] the case $N=2$. In all these works the range of $b$ is the same one where local well-posedness was obtained.

Motivated by the aforementioned papers, our main interest in this paper is to prove similar results for the INLS_a equation in $H^{1}_{a}(\mathbb{R}^{N})$. The well-posedness theory was already studied by Suzuki [25]. Using the energy method, the author showed that, if¹¹1It is worth mentioning that in [25] the author considered (1.1) with $a=-\frac{(N-2)^{2}}{4}$, the critical coefficient. The proof for the case $a>\frac{(N-2)^{2}}{4}$ is an immediate consequence of the previous one.

then (1.1) is locally well-posed in $H^{1}_{a}(\mathbb{R}^{N})$ satisfying $u\in C\left([0,T);H^{1}(\mathbb{R}^{N})\right)\cap C^{1}\left([0,T);H^{-1}(\mathbb{R}^{N})\right)$ for some $T>0$. It was also proved that any local solution of the IVP (1.1) with $u_{0}\in H^{1}_{a}(\mathbb{R}^{N})$ extends globally in time if either $\lambda=-1$ (defocusing case) or $0<\alpha<\tfrac{4-2b}{N}$ for $\lambda=1$ (focusing, $L^{2}$-subcritical case).

Our first goal here is to study global existence and blow-up in $H^{1}(\mathbb{R}^{N})$ for $\lambda=1$ for both the $L^{2}$-critical and the intercritical ($L^{2}$-supercriticial and $H^{1}$-subcritical) case. For that, we apply a sharp Gagliardo-Nirenberg type estimate. In order to do so, we prove the existence of a ground state.

Using the variational analysis associated to Proposition 1.1, we have thresholds for global existence and blow-up.

The main tools for proving Theorems 1.2 and 1.3 are the coercivity given by the variational analysis, and the virial identities. The main difficulty is to appropriately control the error terms appearing when one truncates the virial identity in the radial case. The $L^{2}$-critical case is especially delicate, since the criticality gives us less room for error terms.

Note that the local well-posedness showed in [25] as well as the global results, Theorems 1.2a) and 1.3a) ensure the existence of solutions to (1.1). However we do not know whether or not the solutions satisfy $u\in L^{q}(I;H^{1,r}_{a})$ for any $L^{2}$-admissible pair ($q,r$), which is a key property to study other problems such as scattering for example. To obtain this extra information and working towards the proof of scattering in a next work, we establish the local and global well-posedness for (1.1) via Kato’s method, which is based on the contraction mapping principle and the Strichartz estimates. We start with the local theory for the energy-subcritical case.

New challenges and technical obstructions appear with the presence of the functions $|x|^{-2}$ and $|x|^{-b}$ in (INLS_a), related especially to the problem of equivalence of Sobolev spaces. This leads us to impose some technical restrictions on the parameters $\alpha$, $b$ and $a$, given in (1.12).

As an immediately consequence we obtain the following result.

It is worth mentioning that Corollary 1.6 (ii) can be seen as an extension of a local result by Killip-Murphy-Visan-Zheng [21] to the INLS_a model.

In the sequel we establish small data global results in $H^{1}_{a}(\mathbb{R}^{N})$, for $N\geq 3$ and $\tfrac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}$. Since we use the Strichartz estimates (see Section 2) we show global well-posedness for radial and non-radial initial data. Here, $e^{-it\mathcal{L}_{a}}u_{0}$ denotes the solution to the linear problem associated to (1.1) and the Strichartz norm $\|\cdot\|_{S(\dot{H}^{s_{c}})}$ is defined in Section 2.1.

In Theorem 1.8, when $N=3$, we have an extra restriction on $\alpha$, namely $\alpha<3-2b$. To reach $\alpha<4-2b$, we need to restrict the parameter $b$. This restriction comes from the need of $\alpha>1$ in the fixed point argument. To this end, we use the norm $\|u\|_{L^{\bar{a}}_{t}L^{\bar{r}}_{x}}$ denoted by

where ($\bar{a},\bar{r})$ only satisfies $\bar{r}\geq 2$ and the relation of $\dot{H}^{s_{c}}$-admissible pair, i.e., $\frac{2}{\bar{a}}=\frac{N}{2}-\frac{N}{\bar{r}}-s_{c}$ and not the remaining conditions (see Section 2).

We remark that Theorem 1.9 holds for general initial data. On the other hand, Theorem 1.9-(ii) shows global well posedness in the full intercritical regime, however with $b<\frac{1}{2}$. The gap $\frac{1}{2}\leq b<\frac{3}{2}$ is still an open problem. Moreover, in the particular case³³3The case $\frac{4-2b}{3}<\alpha<3-2b$ was obtained in Theorem 1.8. $3-2b\leq\alpha<4-2b$, we have a better range for $b$ than in (ii).

The next result holds for non-radial data and $a<0$, however only for dimensions $N=3,4,5$. Here, we also use the norm $\|\cdot\|_{\widetilde{S}(\dot{H}^{s_{c}})}$.

As mentioned above the main tool to show the local and global well-posedness is the Fixed Point Theorem, which is based on the Strichartz estimates. Similarly as in the local theory the main difficulty here is to look for admissible pairs to establish the equivalence of Sobolev spaces, mainly when $a<0$.

Once global results are proved, the natural route is to study the asymptotic behavior of such global solutions as $t\rightarrow\pm\infty$. We show that our solutions scatter to a solution of the linear problem in $H^{1}_{a}(\mathbb{R}^{N})$. In addition, we construct the wave operator associated to eq. (1.1). This is the reciprocal problem of the scattering theory, which consists in constructing a solution with a prescribed scattering state.

The rest of the paper is organized as follows. In Section 2, we introduce some notations and give a review of the Strichartz estimates. In Section 3 we discuss the existence of a ground state and establish global existence as well as blow up in $H^{1}_{a}(\mathbb{R}^{N})$, for $L^{2}$-critical and intercritical cases. In Section 4 we study the local and global well-posedness applying the contraction mapping principle. Finally, in Section 5 we prove Theorem 1.12.

In this section, we introduce the notation used throughout the paper and list some useful results. We use $C$ to denote various constants that may vary line by line. If $a$ and $b$ be positive real numbers, the notation $a\lesssim b$ means that there exists a positive constant $C>0$ such that⁴⁴4The constant $C$ may depend on parameters, such as the dimension $N$, as well on a priori estimates on the solution, but never on the solution itself or on time $a\leq Cb$. The notation $a\sim b$ means $a\lesssim b$ and $b\lesssim a$. Given a real number $r$, we use $r^{\pm}$ to denote $r\pm\varepsilon$ for some $\varepsilon>0$ sufficiently small. For a subset $A\subset\mathbb{R}^{N}$, its complement is denoted by $A^{C}=\mathbb{R}^{N}\backslash A$ and the characteristic function $\chi_{A}$ denotes the function that has value $1$ at points of $A$ and $0$ at points of $A^{C}$. Given $x,y\in\mathbb{R}^{N}$, $x\cdot y$ denotes the usual inner product of $x$ and $y$ in $\mathbb{R}^{N}$.

The norm in the Sobolev spaces $\dot{H}^{s,r}=\dot{H}^{s,r}(\mathbb{R}^{N})$ and $H^{s,r}=H^{s,r}(\mathbb{R}^{N})$, are defined by

where $D^{s}f:=\sqrt{-\Delta}^{s}f=(|\xi|^{s}\widehat{f})^{\vee}$ and $\langle\cdot\rangle=(1+|\cdot|^{2})^{\frac{1}{2}}$. If $r=2$ we denote $H^{s,2}$ and $\dot{H}^{s,2}$ simply by $H^{s}$ and $\dot{H}^{s}$, respectively. Similarly, we define Sobolev spaces $\dot{H}_{a}^{s,r}$ and $H_{a}^{s,r}$ associated to $\mathcal{L}_{a}$ by the closure of $\mathbb{C}^{\infty}_{0}(\mathbb{R}^{N}\backslash\{0\})$ under the norms

We abbreviate $\dot{H}_{a}^{s}(\mathbb{R}^{N})=\dot{H}_{a}^{s,2}(\mathbb{R}^{N}))$ and $H_{a}^{s}(\mathbb{R}^{N})=H_{a}^{s,2}(\mathbb{R}^{N})$. Note that, by the sharp Hardy inequality, one has

We also define, for $1\leq p<\infty$, the weighted Sobolev space $L^{p}_{b}=L^{p}_{b}(\mathbb{R}^{N})=\{f:\;\|f\|_{p,b}<+\infty\}$, where

Let $q,r>0$, $s\in\mathbb{R}$, and $I\subset\mathbb{R}$ an interval; the mixed norms in the spaces $L^{q}_{I}L^{r}_{x}$ and $L^{q}_{I}H^{s}_{x}$ of a function $f=f(t,x)$ are defined as

with the usual modifications if either $q=\infty$ or $r=\infty$. When the space 99+-6integration is restricted to a subset $A\subset\mathbb{R}^{N}$ then the mixed norm will be denoted by $\|f\|_{L_{I}^{q}L^{r}_{x}(A)}$. Moreover, if $I=\mathbb{R}$ we shall use the notations $\|f\|_{L_{t}^{q}L^{r}_{x}}$ and $\|f\|_{L_{t}^{q}H^{s}_{x}}$.

Next, we recall some important inequalities. To state the estimates below, it is useful to introduce the parameter

The next lemma implies that the Strichartz estimates (Lemma 2.6) hold.