Novel antimicrobial strategies against infections

1.   Introduction

This article discusses the Cauchy problem for the following inhomogeneous nonlinear Schrödinger equation (NLS) with inverse-square potential:

where $N\geq3$, $0 < T \leq \infty$, $\varphi : [0, T) \times \mathbb{R}^{N} \rightarrow \mathbb{C}$, $K(x)\in C^{1}(\mathbb{R}^{N}, \mathbb{R})$, $2 < p\leq p^{*} = \frac{2N}{N-2}$ and $0 < c\leq c^{*}$, where $c^{*} = \frac{(N-2)^{2}}{4}$ represents the sharp constant of the following Hardy's inequality:

The model of inhomogeneous NLS (1.1) with inverse-square potential can be applied to a variety of physical environments, such as black hole solutions of Einstein's equations or quantum field equations (see e.g., ^[1,2,3]) and quantum gas theory (see e.g., ^[4,5,6]).

In the present work, our interest focuses on the optimal criteria of global existence and finite-time blow-up, as well as the $L^{2}$-concentration property and the dynamics of blow-up solutions for Eq (1.1), which are pursued in both mathematics and physics.

Before going further, we recall some existing results. When $c = 0$, Eq (1.1) becomes the following common inhomogeneous NLS:

which is widely used in fields such as quantum mechanics, nonlinear optics and Bose-Einstein condensation. In the past several decades, this kind of NLS has garnered a great deal of interest. Under the condition of $K(x) = 1$ in Eq (1.2), Weinstein ^[7] not only applied the ground-state solution to the scalar equation

to establish the sharp criterion of global and blow-up solutions, he also showed the existence of the unstable standing wave solutions in the $L^{2}$-critical case ($p = 2+\frac{4}{N}$). Merle, in ^[8], proved the mass concentration of explosive solutions and classified the minimal-mass blow-up solutions on non-radial data in the $L^{2}$-critical setting by utilizing the concentration lemma and pseudo-conformal transformation, together with the variational characterization of the ground-state solution to Eq (1.3). In ^[9], Hmidi and Keraani proposed a refined compactness lemma by applying the profile decomposition to bounded sequences in $H^{1}(\mathbb{R}^{N})$. In view of the obtained compactness result, they also gave the simpler proofs to the concentration property of solutions blowing up in finite time, the limiting profile and determination of the minimal blow-up solutions to the homogeneous $L^{2}$-critical NLS.

Under the condition that $K(x)\neq constant$ and satisfies some appropriate assumptions, Merle ^[10] explored, in detail, the existence or nonexistence for the minimal-mass blow-up solutions, as well as the blow-up dynamical properties of solutions to Eq (1.2). Shu and Zhang ^[11] derived a sharp condition for the existence of a global solution to Eq (1.2) with the mass critical exponent $p = 2+\frac{4}{N}$ by constructing some cross-invariant manifolds and variational problems. It is also worth noting that, for $p = 2+\frac{4}{N-2}$ and $K(x)\neq constant$ satisfying $(A_{1})$ – $(A_{2})$ (see Section 2), Liu ^[12] established a new sharp criterion for the explosive solutions to the $H^{1}$-critical inhomogeneous Eq (1.2) with $p = 2+\frac{4}{N-2}$ for the non-radial case by using the potential well method. These studies are strongly dependent on the hypotheses imposed on the inhomogeneous coefficient $K(x)$. Then, a natural problem arises: Are these results valid for the inhomogeneous NLS (1.1) with inverse-square potential ($c\neq0$) and variable coefficient $K(x)\neq constant$? It is one of our starting points to study problem (1.1) in the current article.

For the case that $c\neq 0$ and $K(x) = 1$, Eq (1.1) corresponds to the homogeneous NLS with inverse-square potential, which has been extensively discussed due to the singular property of inverse-square potential $c|x|^{-2}$. In ^[13,14], the authors researched the global scattering and blow-up problems for the focusing and defocusing NLS in the intercritical and energy-critical settings, respectively. By making full use of the ground state to the elliptic equation

Dinh ^[15] showed the finite-time blow-up and global existence of radially symmetric and non-radial solutions of Eq (1.1) in the mass-critical and mass-supercritical settings with $c < 0$ or $0 < c < c^{*}$, as well as the criterion of blow-up solutions for the energy-critical case with $c\neq0$ and $c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$, respectively. By replacing the power-type nonlinearity by $-(I_{\alpha}*|\varphi|^{p})|\varphi|^{p-2}\varphi$ in Eq (1.1), in light of Dinh ^[15], Li ^[16] investigated the criteria of global existence versus blow-up for non-radial and radial solutions to the $L^{2}$-critical and $L^{2}$-supercritical Choquard equation in the cases of $0 < c < c^{*}$ and $c < 0$, respectively. Under the condition that $N\geq3$, $p = 2+\frac{4}{N}$ and $0 < c < c^{*}$ in Eq (1.1), Csobo and Genoud ^[17] showed that the ground-state solution to the mass critical Eq (1.4) exists, and they obtained an optimal condition for global existence. In addition, the ground-state solution of Eq (1.4) and pseudo-conformal transformation were applied to classify the explosive solutions with minimal-mass for Eq (1.1) in ^[17], and a detailed characterization of minimal blow-up solutions was given via the variational characteristic of the ground-state solution to Eq (1.4) and the concentration compactness principle related to the Hardy functional. For the critical case of $c = c^{*}$ in Eq (1.1), Mukherjee et al. ^[18] was the first to verify the existence and uniqueness of the ground-state solution to Eq (1.4) for $2 < p < p^{*}$, which was applied to establish the criteria of blow-up versus global existence dichotomy for the homogeneous Eq (1.1). Moreover, by taking advantage of the variational characterization to the ground-state solution of Eq (1.4) and the corresponding concentration compactness principle, they studied the mass concentration phenomenon of explosive solutions and gave a complete characterization of the minimal-mass blow-up solutions. Regarding the case that $p = 2+\frac{4}{N}$ and $0 < c < c^{*}$, Bensouilah ^[19] proposed a refined compactness lemma related to the Hardy functional by using the profile decomposition technique in $H^{1}(\mathbb{R}^{N})$, and they applied it to investigate the $L^{2}$-concentration property of solutions blowing up at time $0 < T < \infty$. Based on ^[17] and ^[19], Pan and Zhang ^[20] demonstrated the accurate $L^{2}$-concentration property for the explosive solutions with a minimal mass by using the variational characterization to the ground state of Eq (1.4) and the compactness lemma proposed by ^[19]. For $0 < c < c^{*}$ or $c < 0$ in Eq (1.1), the authors of ^[21,22] studied the stability and strong instability of standing waves for different values of $p$. For the homogeneous NLS including combined nonlinearities and inverse-square potential, Xia ^[23], Cao ^[24] and Li and Zou ^[25] researched the global existence and scattering, the existence of stable standing waves or the existence and properties of normalized solutions, respectively. Regarding the standing waves, Goubet and Manoubi ^[26] researched the existence and orbital stability of standing waves for a class of NLS involving a discontinuous dispersion. Zuo et al. ^[27] applied a variational approach based on the scaling function method to investigate the existence of normalized solutions for a kind of fractional NLS with bounded parametric potential; the solutions have attracted widespread attention due to their important applications in many settings, such as physics.

For $K(x) = \lambda|x|^{-b}$ with $\lambda = \pm1$, $0 < b < 2$ and $0 < c < c^{*}$, Campos and Guzmán ^[28] and An et al. ^[29] considered the blow-up and global solutions to Eq (1.1). Pan and Zhang ^[30] studied the existence of a ground-state solution to the corresponding $L^{2}$-critical inhomogeneous elliptic equation with $K(x) = |x|^{-b}$, and they proved the uniqueness of the minimal-mass blow-up solutions by using the concentration compactness principle, with respect to the Hardy functional and inhomogeneous nonlinearity, as well as variational characterization to the ground state. To the best of our knowledge, there is no literature concerning the inhomogeneous Eq (1.1) that includes inverse-square potential and the general $C^{1}$ variable coefficient $K(x)$, which has significant differences with the cases of ^{[15,17,18,19,20]}. Therefore, it is particularly meaningful for us to study Eq (1.1).

Motivated by the works mentioned above, the aims of this paper are to gain the criteria for the existence of global or finite-time blow-up solutions to Eq (1.1) in the $L^{2}$-critical, $L^{2}$-supercritical cases and the sharp criterion of blow-up in the energy-critical case, as well as the blow-up dynamics of solutions in finite time. The main difficulty stems from the presence of inverse-square potential $c|x|^{-2}$ and variable coefficient $K(x)$, leading to the loss of pseudo-conformal symmetry and scaling invariance, which play a vital role in the research on the blow-up dynamics; see ^[9,18] for example. To overcome the difficulty, we utilize the unique ground-state solution of Eq (3.1) to establish the global existence and blow-up results to the mass-critical and mass-supercritical inhomogeneous NLS and apply the ground state to characterize the blow-up dynamic behavior of solutions to Eq (1.1). To be more precise, enlightened by ^[15,18], we first obtain some sharp thresholds for global existence and finite-time blow-up in the $L^{2}$-critical and $L^{2}$-supercritical cases for $0 < c\leq c^{*}$ in terms of the ground state for Eq (3.1). It is worth mentioning that, in this work, for $0 < c < c^{*}$ and $p = 2+\frac{4}{N}$, the argument regarding the existence of explosive solutions is established through scaling techniques, which differs from the methods of ^[15,18]. Then, under the assumptions $(A_{1})$ – $(A_{2})$ on the inhomogeneous coefficient $K(x)$ (see Section 2), following the ideas of ^[12] and ^[15], we derive the sharp criterion of blow-up in the energy-critical case with $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$ by utilizing the potential well method and the sharp Sobolev constant. Finally, in light of ^[10,19,20], we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $L^{2}$-critical setting for $0 < c < c^{*}$. The main ingredients we use in the proofs of the dynamics are the variational characterization of the ground state for Eq (3.1), scaling techniques and a refined compactness lemma proposed by Bensouilah ^[19]. Our results generalize and supplement the results of ^{[12,15,18,19,20]}.

The remaining parts of the present article is structured as follows. Section 2 gives some notations and important hypotheses, as well as some useful lemmas. Section 3 considers the criteria for the global existence and finite-time blow-up of Eq (1.1) in the $L^{2}$-critical and $L^{2}$-supercritical cases, as well as the sharp blow-up criterion in the energy-critical case, respectively. The last section focuses on the blow-up dynamics of solutions in the $L^{2}$-critical setting with $0 < c < c^{*}$.

2.   Notations and preliminaries

To simplify the symbols, we use the abbreviation $\int \cdot \ dx$ to represent $\int _{\mathbb{R}^{N}} \cdot \ dx$, and we denote $||\cdot||_{L^{p}}\ (1\leq p < \infty)$ by $||\cdot||_{p}$. $C$ represents a positive constant, which may vary from line to line.

Hereafter, we assume that the inhomogeneous coefficient $K(x)$ satisfies some of the following hypotheses: there exist $K_{2}\geq K_{1} > 0$ such that

$(A_{1})\ \forall x\in \mathbb{R}^{N}, \ K_{1} = inf_{x\in\mathbb{R}^{N}}K(x)\leq K(x)\leq sup_{x\in \mathbb {R}^{N}}K(x) = K_{2} < \infty;$

$(A_{2})\ \forall x\in \mathbb{R}^{N}, \; x\cdot\nabla K(x)\leq0 \ and \ |x\cdot\nabla K(x)| \leq C;$

$(A_{3})\ there\; is\; x_{0}\in\mathbb{R}^{N}\; satisfying\; that\; K (x_{0}) = K_{2}.$

In accordance with Dinh ^[15] and Okazawa et al. ^[31], we have the following argument regarding the local well-posedness of solutions to Eq (1.1).

Proposition 2.1. Let $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$, $2 < p < p^{*}$ and $0 < c\leq c^{*}$ or $p = p^{*}$ and $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$, and assume that $(A_{1})$ holds; then, for $T\in(0, \infty]$ (maximal existence time), the unique solution $\varphi(t, x)\in C([0, T), H^{1}(\mathbb{R}^{N}))$ for Eq (1.1) exists. Meanwhile, one has the alternative $T = \infty$ (global existence), or else $T < \infty$ and $\lim_{t\rightarrow T}||\varphi(t, x)||_{H^{1}(\mathbb{R}^{N})}$ $= \infty$ (blow-up). Furthermore, for all $t\in[0, T)$, the solution $\varphi(t, x)$ possesses the conserved quantities of mass and energy as shown below:

Define the Hardy functional as below:

which is of great importance in analyzing the dynamical properties for blow-up solutions. Taking account of the hypothesis on $c$, the semi-norm defined by $H(\varphi)$ on $H^{1}(\mathbb{R}^{N})$ is equivalent to $||\nabla\varphi||_{2}^{2}$. Thus, the solution $\varphi(t, x)$ to Eq (1.1) blows up at $T > 0$ if and only if $\lim_{t\rightarrow T}H(\varphi) = \infty$.

To go further, we review some useful lemmas.

Lemma 2.2. (Hardy-Gagliardo-Nirenberg inequality (^[18,32])) Let $N \geq 3$, $0 < c\leq c^{*}$ and $2 < p < p^{*}$. Then, we obtain

for all $\varphi \in H^{1}(\mathbb{R}^{N})$, with the sharp constant

where $Q(x)$ is the unique radial positive solution to the elliptic equation (1.4).

Lemma 2.3. (^[7]) Assume that $\varphi \in H^{1}(\mathbb{R}^{N})$; then, we get

Lemma 2.4. (Sharp Sobolev embedding (^[33])) Let $N \geq 3$ and $0 < c < c^{*}$; then, we have

where the sharp Sobolev constant $C_{SE}(c)$ is as follows:

3.   Global existence and blow-up of the solutions to Eq (1.1)

In the current section, we are devoted to researching the criteria for the global existence and finite-time blow-up to Eq (1.1) in the $L^{2}$-critical and $L^{2}$-supercritical settings, as well as the sharp threshold of blow-up in the energy-critical case, respectively. As we know, the ground state has a crucial role in the criteria for blow-up versus global existence, and it is the unique positive radially symmetric solution of the following elliptic equation with power nonlinearity and inverse-square potential:

where $K_{2}$ is same as the hypotheses $(A_{1})$ – $(A_{3})$. Furthermore, we will apply the ground-state solution of Eq (3.1) to characterize the dynamical properties of explosive solutions in the next section.

Assume that $Q(x)$ is the unique radial positive solution to Eq (1.4); by the scaling transformation $Q_{K_{2}}(x) = K_{2}^{-\frac{1}{p-2}}Q(x)$, it is easy to obtain that $Q_{K_{2}}(x)$ is the positive ground state of Eq (3.1). Combining Eqs (1.4) and (3.1), we immediately get the scaling identity and Poho$\breve{z}$aev identities as follows:

Define the following functionals:

where $\sigma = \frac{2N+2p-Np}{Np-2N-4}$ when $2+\frac{4}{N} < p < 2+\frac{4}{N-2}$. From Eqs (2.3), (3.2) and (3.3), one has

and

It follows from Eqs (3.2)–(3.6) that

and

In particular, we have

Now, we consider the virial-type identities, which play a key role in the research of the existence of explosive solutions to Eq (1.1). Let

and for $\varphi(t, x)\in \Sigma$, we introduce the variance functional

Then, we are able to derive the following conclusion.

Proposition 3.1. Assume that $2 < p < p^{*}$ and $0 < c\leq c^{*}$ or $p = p^{*}$ and $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$, and let $\varphi(t, x)$ be a solution of problem (1.1) on $t\in [0, T)$. If $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, then $\varphi(t, x)\in\Sigma$ for any $t\in [0, T)$ and $V(t)$ satisfies the following identities:

and

Proof. Based on the work of Csobo and Genoud ^[17] (see also Cazenave ^[34]), by a formal computation, it is easy to obtain that

and

where

From Eqs (3.10)–(3.12), we claim that Eq (3.9) holds.

From Proposition 3.1 and Lemma 2.3, we get the following sufficient conditions for blow-up.

Corollary 3.2. Assume that $2+\frac{4}{N}\leq p\leq p^{*}$ and $(A_{1})$ and $(A_{2})$ hold; if $\varphi_0\in \Sigma$ and $\varphi_{0}$ meets one of the three conditions below:

Case 1): $E(\varphi_0) < 0$;

Case 2): $E(\varphi_0) = 0$ and $Im\int x\nabla\varphi_0 \bar{\varphi_0}dx < 0$;

Case 3): $E(\varphi_0) > 0$ and $Im\int x\nabla\varphi_0 \bar{\varphi_0}dx+ (2V(0)E(\varphi_0))^{\frac{1}{2}}\leq0$,

then blow-up of the solution $\varphi(t, x)$ to Eq (1.1) occurs in finite time.

Proof. Since $K(x)$ satisfies $(A_{1})$ and $(A_{2})$, using Proposition 3.1, we have

Thus

Then, for each of the cases 1, 2 or 3, one can deduce that $0 < T < +\infty$ must exist and satisfy

This, together with Lemma 2.3, implies that

Therefore, the explosion of the solution $\varphi(t, x)$ to Eq (1.1) happens in the time period of $0 < T < +\infty$.

3.1.   $L^{2}$-critical case

The first argument of our work is about the global existence and blow-up of Eq (1.1) in the $L^{2}$-critical setting (i.e., $p = 2 + \frac{4}{N}$).

Theorem 3.3. Assume that $p = 2 + \frac{4}{N}$ and $\varphi_{0} \in H^{1}(\mathbb{R}^{N})$. Let $Q_{K_{2}}(x)$ be the positive ground state of Eq (3.1).

(i) Global existence: Assume that $(A_{1})$ holds true. If $0 < c\leq c^{*}$ and $\| \varphi_{0} \|_{2} < \| Q_{K_{2}} \|_{2}$, then the solution $\varphi(t, x)$ of Eq (1.1) exists globally in $t \in [0, +\infty)$.

(ii) Blow-up: Assume that $(A_{1})$ and $(A_{2})$ hold.

(a) Then, for $0 < c < c^{*}$, any $\lambda > 0$ and any real constant $C_{1}$ satisfying that $|C_{1}|\geq(\frac{K_{2}}{K_{1}})^{\frac{N}{4}}\geq1$,

there exists $\varphi_{0} = C_{1}\lambda^{\frac{N}{2}}Q_{K_{2}}(\lambda x)\in\Sigma$ such that

and blow-up of the corresponding solution $\varphi(t, x)$ to problem (1.1) occurs in $0 < T < +\infty$.

(b) For $c = c^{*}$, if $E(\varphi_{0}) < 0$ and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, then the blow-up of the solution $\varphi(t, x)$ for Eq (1.1)

happens in the time period of $0 < T < +\infty$.

Proof. (ⅰ) Since $p = 2 + \frac{4}{N}$, we have that $p\theta = 2$ and $pC_{HGN}^{p} = 2\|Q\|_{2}^{\frac{4}{N}}$. Thus, from Eq (2.2), $(A_{1})$, Eqs (2.3) and (2.1), we have the following estimate:

Since $\|\varphi_{0}\|_{2} < \|Q_{K_{2}}\|_{2}$, $H(\varphi)$ is bounded uniformly for $t\in [0, +\infty)$. From Proposition 2.1, we claim that $\varphi(t, x)$ must be a global solution.

Next, we prove part (ⅱ) of Theorem 3.3. For $0 < c < c^{*}$, let

for any $\lambda > 0$, and $C_{1}\in \mathbb{R}$ will be determined later. Based on the scaling arguments, one has that

Take

then, we have that $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$. Indeed, according to Bensouilah-Dinh-Zhu's conclusion in ^[21], one has that $Q(x)\in H^{1}(\mathbb{R}^{N})$ and $|x|Q(x)\in L^{2}(\mathbb{R}^{N})$. Thus, we obtain that $Q_{K_{2}}(x)\in H^{1}(\mathbb{R}^{N})$ and $|x|Q_{K_{2}}(x)\in L^{2}(\mathbb{R}^{N})$, which yields that $\varphi_{0} = C_{1}\lambda^{\frac{N}{2}}Q_{K_{2}}(\lambda x)\in \Sigma$. Moreover, it follows from Eq (3.13) that

From Eq (2.2), $(A_{1})$, Eqs (3.14)–(3.16) and the Poho$\breve{z}$aev identities given by Eq (3.3), we have

where the last inequality is based on the fact that $p = 2+\frac{4}{N}$ and $|C_{1}|\geq(\frac{K_{2}}{K_{1}})^{\frac{N}{4}}\geq1$. Thus, we infer from Corollary 3.2 that blow-up of the corresponding solution $\varphi(t, x)$ to problem (1.1) occurs in $0 < T < +\infty$.

For $c = c^{*}$, if $E(\varphi_{0}) < 0$ and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, then, by Corollary 3.2, it is easy to derive that blow-up of the solution $\varphi(t, x)$ for Eq (1.1) happens in the time period of $0 < T < +\infty$.

3.2.   $L^{2}$-supercritical case

Then, we analyze the $L^{2}$-supercritical case (i.e., $2+\frac{4}{N} < p < p^{*} = 2+\frac{4}{N-2}$). For this case, we obtain the threshold for global existence as shown below.

Theorem 3.4. Suppose that $N\geq 3$, $0 < c\leq c^{*}$ and $2+\frac{4}{N} < p < p*$. Let $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ and $\varphi(t, x)$ be the corresponding solution of Eq (1.1). Suppose that

(i) Global existence: Assume that $(A_{1})$ holds. If

then the global solution $\varphi(t, x)$ to Eq (1.1) exists. In addition,

(ii) Blow-up: Assume that $(A_{1})$ – $(A_{2})$ hold. If

and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, then the finite-time blow-up solution $\varphi(t, x)$ of Eq (1.1) exists and

for any $t\in [0, T)$. Furthermore, the finite-time blow-up result still holds true if we assume that $E(\varphi_{0}) < 0$, in place of Eqs (3.17) and (3.19).

Proof. (ⅰ) From Eq (2.2), $(A_{1})$ and Eq (2.3), one has

where

Enlightened by the idea of ^[35] (the function $f$ in ^[35] differs from ours; however, this change does not make a significant difference), we will utilize an important fact that $f$ strictly increases in $[0, G(c)]$ and strictly decreases in $[G(c), \infty)$. Moreover, from Eqs (3.21), (3.7) and (3.8), we get

Thus, combining this with Eqs (3.17), (3.21) and (3.22), we have

From the above inequality, Eq (3.18) and the continuity argument, we deduce that

From Eq (2.1), we obtain the boundedness of $H(\varphi)^{\frac{1}{2}}$, which implies that the solution $\varphi(t, x)$ of Eq (1.1) exists globally.

We next treat part (ⅱ) of Theorem 3.4. For the case that $E(\varphi_{0})\geq 0$, we claim that Eq (3.20) holds. Indeed, from Eqs (3.23) and (3.19) and the continuity argument, one has

which means that Eq (3.20) holds true.

On the other hand, from Eq (3.17) and the continuity argument, we can take $\delta > 0$ small enough such that

which yields that

Applying Eqs (3.21), (3.7) and (3.8) to Eq (3.25), one has that

By making use of the continuity argument again, we deduce from Eq (3.19) that there exists $\delta^{'} > 0$ that relies upon $\delta$ satisfying

Moreover, taking

then, for any $t\in[0, T)$, we claim that

for $\varepsilon > 0$ small enough. In fact, multiplying $LHS$ by $||\varphi||_{2}^{2\sigma}$, we obtain

Utilizing Eqs (2.1), (2.2), (3.24), (3.26) and (3.8), one has

We readily obtain $LHS\leq -C < 0$ by taking $\varepsilon > 0$ small enough. Then, it follows from the virial identity (3.9) and $(A_{2})$ that

This yields that blow-up of the solution $\varphi(t, x)$ must occur in the time period of $0 < T < +\infty$.

The case that $E(\varphi_{0}) < 0$ is easy. By Corollary 3.2, we conclude that the finite-time blow-up solution $\varphi(t, x)$ of Eq (1.1) exists.

3.3.   Energy-critical setting

Finally, the energy-critical setting (i.e., $p = p^{*} = 2+\frac{4}{N-2}$) for problem (1.1) is considered in this subsection. According to Dinh ^[15], taking account the sharp constant (see also Eq (2.4))

then we are able to get the optimal blow-up criterion.

Theorem 3.5. Assume that $p = p^{*}$, $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$, $\varphi_{0} \in H^{1}(\mathbb{R}^{N})$ and $E(\varphi_{0}) < \frac{S^{\frac{N}{2}}}{NK_{2}^\frac{N-2}{2}}$. Let $\varphi(t, x)$ be the corresponding solution of Eq (1.1), defined on $[0, T)\times \mathbb{R}^{N}, \ 0 < T \leq \infty.$

(i) Uniform boundedness: Assume that $K(x)$ satisfies $(A_{1})$. If $H(\varphi_{0}) < \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$, then the solution $\varphi(t, x)$ of Eq (1.1) is bounded in $H^{1}(\mathbb{R}^{N})$ for $t\in[0, T)$ and $H(\varphi(t)) < \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$.

(ii) Blow-up: Assume that $K(x)$ satisfies $(A_{1})$ – $(A_{2})$. If $H(\varphi_{0}) > \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$ and $|x|\varphi_{0}\in L^{2}(\mathbb{R}^{N})$, then there exists $0 < T < \infty$ such that the solution $\varphi(t, x)$ of Eq (1.1) blows up at $T$.

Proof. (ⅰ) Assume that $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ satisfies that $E(\varphi_{0}) < \frac{S^\frac{N}{2}}{NK_{2}^\frac{N-2}{2}}$ and $H(\varphi_{0}) < \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$. We claim that

We demonstrate Eq (3.27) by contradiction. Suppose that there exists $t_{1}\in (0, T)$ such that $H(\varphi(t_{1})) = \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$ by using the continuity of the solution $\varphi(t, x)$ in $H^{1}(\mathbb{R}^{N})$ at time $t$. By Eq (2.2), $(A_{1})$ and Lemma 2.4, we found that

which contradicts the fact that $H(\varphi(t_{1})) = \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$. Therefore, Eq (3.27) holds. This means that the solution $\varphi(t, x)$ is bounded in $H^{1}(\mathbb{R}^{N})$ for $t\in[0, T)$.

Now, we prove part (ⅱ) of Theorem 3.5. Since $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ satisfies that $E(\varphi_{0}) < \frac{S^\frac{N}{2}}{NK_{2}^\frac{N-2}{2}}$ and $H(\varphi_{0}) > \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$. It remains to be proven that

If otherwise, since $\varphi(t, x)$ is continuous with respect to time $t$ in $H^{1}(\mathbb{R}^{N})$, we deduce that there exists $t_{2}\in (0, T)$ satisfying that $H(\varphi(t_{2})) = \frac{S^{\frac{N}{2}}}{K_{2}^{\frac{N-2}{2}}}$. Combining Eq (2.2) with $(A_{1})$, and by Lemma 2.4, one has the following estimate:

which gives a contradiction. Thus, Eq (3.28) holds. Keeping in mind that $p^{*} = 2+\frac{4}{N-2}$ and $\varphi_{0}\in H^{1}(\mathbb{R}^{N})$ with $|x|\varphi_{0} \in L^{2}(\mathbb{R}^{N})$, we have

Then, we infer from Eqs (3.28) and (2.2) that

Inserting Eq (3.29) into $V^{\prime\prime}(t)$, and then from $(A_{2})$, one has the estimate

from which we know that explosion of the solution $\varphi(t, x)$ to Eq (1.1) must happen within the time period of $0 < T < \infty$.

Remark 3.6. (i) Under the conditions that $K(x) = 1$, $c = c^{*}$ and $2 < p < p^{*}$ in Eq (1.1), Mukherjee et al. ^[18] studied the criterion of blow-up versus global existence in the $L^{2}$-critical and $L^{2}$-supercritical cases (see ^[18], Theorem 3); Dinh ^[15] obtained the blow-up and global existence results for Eq (1.1) for non-radial data in the mass-critical and mass-supercritical cases with $0 < c < c^{*}$, as well as the sharp blow-up criterion in the energy-critical case with $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$ (see ^[15], Theorems 1.3, 1.6 and 1.12). Our results (Theorems 3.3–3.5) generalize the results of ^[15,18] to the case of the inhomogeneous NLS involving inverse-square potential and a bounded positive variable coefficient.

(ii) Given $c = 0$, Liu ^[12] verified the sharp existence of non-radial finite-time blow-up solutions to the inhomogeneous Eq (1.2) in the energy-critical case $p = \frac{2N}{N-2}$ (see ^[12], Theorem 1.2). We extend this conclusion to the case of the inhomogeneous NLS with inverse-square potential for $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$ (see Theorem 3.5).

(iii) For $0 < c < \frac{N^{2}+4N}{(N+2)^{2}}c^{*}$ and $p = p^{*}$, we can derive that the global solution of Eq (1.1) exists for some initial value small enough.

4.   Dynamics of blow-up solutions in the $L^{2}$-critical setting

In the present part, we investigate the dynamics of blow-up solutions in the $L^{2}$-critical setting ($p = \; 2+\frac{4}{N}$) with $0 < c < c^{*}$, including the mass concentration phenomenon of blow-up solutions and the dynamical properties of blow-up solutions with a minimal mass for Eq (1.1). To achieve these goals, we first recall a key compactness lemma established by Bensouilah ^[19].

Lemma 4.1. Assume that $\{v_n\}_{n = 1}^{\infty}$ is a bounded sequence in $H^{1}(\mathbb{R}^{N})$ satisfying

Then, there exists $\{x_n\}_{n = 1}^{\infty}\subset \mathbb{R}^{N}$ such that, up to a subsequence,

with $\|U\|_{2}\geq(\frac{N}{N+2})^{\frac{N}{4}}\frac{m^{\frac{N}{2}+1}}{M^{\frac{N}{4}}}\|Q(x)\|_{2}$, where $Q(x)$ is the ground-state solution to Eq (1.4).

With Lemma 4.1 in hand, we are able to obtain the following concentration property of the explosive solutions to Eq (1.1).

Theorem 4.2. ($L^2$-concentration) Assume that $K(x)$ satisfies $(A_{1})$ and $(A_{2})$. Suppose that $\varphi(t, x)$ is a solution to Eq (1.1) blowing up at finite time $T$, and that $s(t)$ is a nonnegative real-valued function on $[0, T)$ such that $s(t)\|\nabla\varphi(t)\|_{2}\rightarrow +\infty$ as $t \rightarrow T$. Then, there exists a function $x(t)\in \mathbb{R}^{N}$ for $t < T$ satisfying

where $Q_{K_{2}}(x)$ denotes the ground state for Eq (3.1).

Proof. Take

Suppose that $\{t_n\}_{n = 1}^{\infty}$ is an arbitrary time sequence satisfying that $t_n \rightarrow T$ as $n \rightarrow \infty$, and denote $\rho_n = \rho(t_n)$ and $v_n(x) = v(t_n, x)$. It follows from Eq (2.1) and the definition of $v_{n}$ that

For $v(x)\in H^{1}(\mathbb{R}^{N})$, we define the functional

From $(A_{1})$ and Eqs (4.3), (2.2) and (4.2), we find that

Thus,

Take $M = H(Q_{K_{2}})$ and $m^{2+\frac{4}{N}} = \frac{p}{2K_{2}}H(Q_{K_{2}})$. Thanks to Lemma 4.1, it is sufficient to demonstrate that there exist $U(x)\in H^{1}(\mathbb{R}^{N})$ and $\{x_n\}_{n = 1}^{\infty}\subset \mathbb{R}^{N}$ such that, up to a subsequence,

with

which leads to

This, together with the lower semi-continuity of the $L^{2}$-norm, yields

Since

there exists $n_{0} > 0$ such that for any $n > n_{0}$, we obtain that $A\rho_n < s(t_n)$. Combining this result and Eq (4.7), one has

which means that

According to the arbitrariness of the sequence $\{t_n\}_{n = 1}^{\infty}$, using the fact that $\|U\|_{2}\geq \|Q_{K_{2}}\|_{2}$, we obtain

For any fixed $t\in[0, T)$, it is simple to infer that the function $g(y): = \int_{|x-y|\leq s(t)}|\varphi(t, x)|^{2}dx$ is continuous on $y\in \mathbb{R}^{N}$ and $\lim_{|y|\rightarrow \infty}g(y) = 0$. Thus, for any $0\leq t < T$, there exists a function $x(t)\in\mathbb{R}^{N}$ that satisfies

Therefore, from Eqs (4.8) and (4.9) we infer that Eq (4.1) holds true.

Corollary 4.3. Suppose that $\varphi(t, x)$ is a solution of Eq (1.1) blowing up within the time period of $0 < T < \infty$. Then, for any $l > 0$, there exists $x(t)\in \mathbb{R}^{N}$ for $0 < t < T$ satisfying that

where $Q_{K_{2}}(x)$ denotes the ground state for Eq (3.1) and $B(x(t), l) = \{x\in\mathbb{R}^{N}||x-x(t)|\leq l\}.$

Now, in terms of Theorem 4.2 and Corollary 4.3, we are concerned with the dynamics of blow-up solutions with the mass $||\varphi_{0}||_{2} = ||Q_{K_{2}}||_{2}$.

Theorem 4.4. (Limiting profile) Let $K(x)$ satisfy $(A_{1})$ – $(A_{2})$ and assume that $||\varphi_{0}||_{2} = ||Q_{K_{2}}||_{2}$. Assume that $\varphi(t, x)$ is a corresponding solution to problem (1.1) blowing up within the time period of $0 < T < \infty$; then, there exist two functions $\theta(t)\in[0, 2\pi)$ and $x(t)\in \mathbb{R}^{N}$ that satisfy

where $Q_{K_{2}}(x)$ denotes the ground state for Eq (3.1) and $\rho(t) = \bigg[\frac{H(Q_{K_{2}})} {H(\varphi)}\bigg]^{\frac{1}{2}}$.

Proof. From Theorem 4.2, we have that $\|U\|_{2}\geq\|Q_{K_2}\|_{2}$ (see Eq (4.6)). Then, by using the assumption that $\|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2}$ and Eq (2.1), we obtain

which implies that

It follows from Eqs (4.1) and (4.10) that

From Eq (2.3), one has

where the last inequality is based on the boundedness of $v_n(x+x_{n})$ in $H^1(\mathbb{R}^{N})$. Thus, Eqs (4.11) and (4.12) give us that

Now, we claim that

Indeed, we deduce from Eqs (4.3) and (4.4) that

Then, from Eqs (4.5), (4.15), (4.13), (2.3) and (4.10), one has

which implies that

From this and Eq (4.5), we get Eq (4.14) and

In summary, we identify the properties of the profile $U$ as below:

from which we deduce that there exist $\theta\in[0, 2\pi)$ and $x_{0}\in\mathbb{R}^{N}$ such that

and

where we utilize the variational characterization to the ground state $Q_{K_2}$. Since the sequence $\{t_n\}_{n = 1}^{\infty}$ is arbitrary and tends to $T$ as $n\rightarrow \infty$, one can infer that there exist two functions $\theta(t)\in[0, 2\pi)$ and $x(t)\in \mathbb{R}^{N}$ such that

with $\rho(t) = \bigg[\frac{H(Q_{K_2})}{H(\varphi)}\bigg]^{\frac{1}{2}}\rightarrow0$ as $t \rightarrow T$.

Theorem 4.5. Assume that $K(x)$ satisfies $(A_{1})$ – $(A_{3})$ and that $\|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2}$. Denote $W = \{x\in \mathbb{R}^{N}|K(x) = K_{2}\}$. Let $\varphi(t, x)$ be the corresponding solution to Eq (1.1) blowing up within the time period of $0 < T < \infty$; then the following holds true:

(i) (Location of $L^2$-concentration point) There exists $x_{0}\in W$ such that

where $Q_{K_{2}}$ is the ground state for Eq (3.1).

(ii) (Blow-up rate) There is a positive constant $C > 0$ that satisfies

Proof. (ⅰ) From Eq (2.1) and $\|\varphi_{0}\|_{2} = \|Q_{K_{2}}\|_{2}$, we deduce that

On the other side, according to Theorem 4.2 and Corollary 4.3, for any $l > 0$, one has that

From Eqs (4.18) and (4.19), we have

which shows that

We shall demonstrate in what follows that there exists $x_{0}\in W$ satisfying that

which implies that

As a matter of fact, for any real-valued function $\theta(x)\in\mathbb{R}^{N}$ and any $\beta\in \mathbb{R}$, from $(A_{1})$ and Eq (2.3), one has the following estimate

where $\theta$ is defined in Lemma 2.2. Here in the last equality, we have used the fact that $||Q_{K_{2}}||_{2} = K_{2}^{-\frac{N}{4}}||Q||_{2}$. Therefore, from Eqs (2.1) and (2.2), for any $\beta\in\mathbb{R}$, we obtain

which implies that

Then, choosing $\theta_{j}(x) = x_{j}$ for $j = 1, 2, \cdot\cdot\cdot, N$ in Eq (4.21), it follows from Eqs (1.1), (2.1) and (2.2) that

Take any two sequences $\{t_n\}_{n = 1}^{\infty}$, $\{t_m\}_{m = 1}^{\infty}\subset[0, T)$ such that $\lim\limits_{n\rightarrow \infty}t_n = \lim\limits_{m\rightarrow \infty}t_m = T$. Thus, for all $j = 1, 2, \cdot\cdot\cdot, N$, from Eq (4.22), we have

which means that

Thus,

Let $x_{0} = \frac{\lim\limits_{t\rightarrow T}\int|\varphi(t, x)|^{2}xdx}{\|Q_{K_{2}}\|_{2}^{2}}$; then, one has that $x_{0}\in \mathbb{R}^{N}$ and

On the other hand, notice that $p = 2+\frac{4}{N}$; then, from Proposition 3.1 and $(A_{2})$, we have

Therefore, there is a positive constant $c_{0} > 0$ satisfying that

Thus, one has the following estimate:

From Eq (4.20), one has

We derive from Eq (4.25) that

Combining Eq (4.24) with Eq (4.26), one has

Thus, we have the following for any $l_0 > 0$:

Therefore, for any $\varepsilon > 0$, there exists a large enough $l_0 = l_0(\varepsilon) > 0$ satisfying that

Then, using Eqs (4.27) and (4.20), we infer that, for any $\varepsilon > 0$,

Combining Eqs (4.23) and (4.28), we immediately get

Thus, there exists $x_{0}\in \mathbb{R}^{N}$ (see Eq (4.23)) such that

Now, we claim that $x_{0}\in W = \{x\in\mathbb{R}^{N}|K(x) = K_{2}\}$. Namely, $x_{0}$ is a global maximal point of $K(x)$. Indeed, since $K(x)\in C^{1}$, we deduce from Corollary 4.3 and Eq (4.29) that

where $Q_{K(x(t))} = K(x(t))^{-\frac{N}{4}}Q(x(t))$. This implies that $K(x_{0}) = K_{2}$. Thus,

Therefore, it is clear that Eq (4.16) holds true.

(ⅱ) Choosing $\theta(x) = |x-x_{0}|^{2}$ in Eq (4.21), we get

which indicates that

Therefore, for any $t\in [0, T)$, by integrating on both sides of this inequality from $t$ to $T$, we derive

Based on the uncertainty principle and Hölder's inequality, we obtain

which means that

Therefore, the conclusion Eq (4.17) holds.

Remark 4.6. (i) Given $K(x) = 1$ and $0 < c < c^{*}$, Bensouilah ^[19] verified the $L^{2}$-concentration of explosive solutions to Eq (1.1) in the $L^{2}$-critical case (see ^[19], Theorem 1). Furthermore, Pan and Zhang ^[20] obtained the precise concentration behavior for blow-up solutions with a critical mass (see ^[20], Theorem 1.1). Our conclusions generalize and supplement the corresponding results of ^[19,20] for the inhomogeneous NLS with inverse-square potential and the coefficient $0 < K(x)\neq constant$ (see Theorems 4.2, 4.4 and 4.5).

(ii) For the case that $c = 0$, $K(x)\neq constant$ and $(A_{1})$ – $(A_{3})$ and other proper conditions are satisfied, the mass concentration property of blow-up solutions and the dynamical behaviors of the $L^{2}$-minimal blow-up solutions have been derived (see ^[10], Theorems 1 and 2). Our results extend the results of ^[10] to the inhomogeneous NLS involving inverse-square potential for $0 < c < c^{*}$ (see Theorems 4.2 and 4.5).

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

This work was partially supported by the Jiangxi Provincial Natural Science Foundation (Grant Nos. 20212BAB211006 and 20224BAB201005) and National Natural Science Foundation of China (Grant No. 11761032). The authors are also greatly thankful to the referees and editors for their helpful comments and advice leading to the improvement of this manuscript.

Conflict of interest

The authors make the declaration that there are no competing interests existing.