L² Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

Abstract

This article studies the Schrödinger equation with an inhomogeneous combined term $i{\partial }_{t}u-{(-\Delta )}^{s}u+{\lambda }_1{|x|}^{-b}{|u|}^{p}u+{\lambda }_2{|u|}^{q}u=0,$ where $s\in (\frac{1}{2},1),{\lambda }_1,{\lambda }_2=\pm 1,0<b<\{2s,N\}$ and $p,q>0$. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters $p,q,{\lambda }_1$ and ${\lambda }_2$ have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the $L^2$ concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

1. Introduction

We consider the Cauchy problem for the fractional Schrödinger equation with a combined non-linearity

$$\left\{\begin{array}{c}i{\partial }_{t}u-{(-\Delta )}^{s}u+{\lambda }_1{|x|}^{-b}{|u|}^{p}u+{\lambda }_2{|u|}^{q}u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,\hfill \\ u(0,x)=u_0(x).\hfill \end{array}\right.$$

(1)

Here and hereafter $N\ge 2$, ${\lambda }_i\in \{\pm 1\}(i=1,2)$ refers to the attractive or repulsive regime. $b\ne 0$ provides an unbounded inhomogeneous term ${|·|}^{-b}$. The exponents of the source terms are $p,q>0$ and $u:=u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}$. The fractional Laplacian operator is defined using the Fourier transform, as follows

$$\mathcal{F}[{(-\Delta )}^s·]:={|·|}^{2s}\mathcal{F}·,s\in (0,1).$$

The fractional Schrödinger problem was first discovered by Laskin [1,2] as a result of extending the Feynmann path integral, from the Brownian-like to L$\acute{e}$vy-like quantum mechanical paths. It also appears in the continuum limit of discrete models with long-range interactions (see [3]) and in the description of Boson stars as well as in water wave dynamics (see [4,5]). In nonlinear optics and plasma physics, the Cauchy problem for the NLS with the inhomogeneous nonlinearity model the beam propagation in an inhomogeneous medium (see [6]). From a mathematical point of view, the problem (1) has no scaling invariance. But the term ${|u|}^{\frac{4s}{N}}u$ is the $L^2$-critical nonlinearity due to the fact that the $L^2$-norm and (1) without the inhomogeneous term are invariant under the scaling symmetry ${\lambda }^{\frac{N}{2}}u({\lambda }^{2s}t,\lambda x)$. The term ${|x|}^{-b}{|u|}^{\frac{4s-2b}{N}}u$ is the $L^2$-critical nonlinearity because the $L^2$-norm and (1) are invariant under the scaling symmetry ${\lambda }^{\frac{N}{2}}u({\lambda }^{2s}t,\lambda x)$ when ${\lambda }_2=0$. For simplification, the term ${|u|}^{\frac{4s}{N}}u$ is called the $L^2$-critical nonlinearity when $p<\frac{4s-2b}{N}$ and the term ${|x|}^{-b}{|u|}^{\frac{4s-2b}{N}}u$ is also called the $L^2$-critical nonlinearity when $q<\frac{4s}{N}$.

Over the past decade, there has been a great deal of interest in studying the fractional Schrödinger problem

$$\begin{array}{c}i{\partial }_{t}u-{(-\Delta )}^{s}u+f(u)=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N.\hfill \end{array}$$

(2)

For the well-posedness of (2), we refer the reader to [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and the references therein. For the $L^2$-critical case, a series of studies dealing with problem (2), we refer to [22,23,24,25,26,27] and the references therein. Recently, in paper [28], we obtained a sharp threshold of global existence versus finite time blow-up dichotomy for mass-super-critical and energy sub-critical radial solutions for (2) when $f(u)=\pm {|x|}^{-b}{|u|}^{p}u\pm {|u|}^{q}u$ in conditions of both radial initial data and non-initial data. In addition, we also investigated the decay of global solutions by Morawetz estimate for radial initial data. For the limiting profile of infinite time blow-up solutions at the blow-up time, in [25,26,29,30,31,32], the authors investigated the $L^2$ concentration phenomenon for different models.

Inspired by the above literature, in this paper, we study the limiting profile of the finite time blow-up solutions at the blow-up time for problem (1). Specifically, we focus on the nonexistence of a strong limit for the non-radial initial data and $L^2$ concentration phenomenon for the radial initial data. As two different nonlinear terms are included, the limiting profile of the finite time blow-up solutions at the blow-up time is different and interesting. For $p<\frac{4s-2b}{N},{\lambda }_2=1$ and $q=\frac{4s}{N}$, we investigate the nonexistence of a strong limit for the non-radial initial data. The difficulty in dealing with this problem stems from the effect of singular coefficient ${|x|}^{-b}$, which is more complex than the classical equation (see [26]). For the radial initial data, we can obtain better results, i.e., the $L^2$ concentration phenomenon, which can be proven by the profile decomposition. For $p=\frac{4s-2b}{N},{\lambda }_1=1$ and $q<\frac{4s}{N}$, we find that the nonexistence of a strong limit for the non-radial initial data is the same as the first case, but the $L^2$ concentration phenomenon for the radial initial data has changed. This is attributed to the effect of the singular coefficient ${|x|}^{-b}$. We cannot use the profile decomposition and have to find some new estimates to deal with the problem. As a result, we only find that the mass of the solution is concentrated near the origin. These results fully illustrate the effect of the inhomogeneous term on the problem. Our main results can be represented in the following

Table 1. Summary of the limiting behavior of the solution.

${\mathit{u}}_0$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	$\mathit{p},\mathit{q}$	Result
$\mathrm{non}–\mathrm{radial}$	$\pm 1$	1	$p<\frac{4s-2b}{N},q=\frac{4s}{N}$	$\mathrm{nonexistence} \mathrm{of} a \mathrm{strong} \mathrm{limit}$
$\mathrm{radial}$	$\pm 1$	1	$p<\frac{4s-2b}{N},q=\frac{4s}{N}$	$L^2\mathrm{concentration}$
$\mathrm{non}–\mathrm{radial}$	1	$\pm 1$	$p=\frac{4s-2b}{N},q<\frac{4s}{N}$	$\mathrm{nonexistence} \mathrm{of} a \mathrm{strong} \mathrm{limit}$
$\mathrm{radial}$	1	$\pm 1$	$p=\frac{4s-2b}{N},q<\frac{4s}{N}$	$L^2\mathrm{concentration} \mathrm{near} \mathrm{the} \mathrm{origin}$

.

The plan of this work is as follows: in the Section 2 we provide some local well-posedness results, a Gagliardo–Nirenberg inequality, and a profile decomposition of the bounded sequence in $H^s$. In Section 3, we provide the proof of nonexistence of a strong limit when $q=\frac{4s}{N}$ for a non-radial solution. For a radial solution, using a compactness lemma and profile decomposition, we prove the $L^2$ concentration phenomenon in Section 4. And when $p=\frac{4s-2b}{N}$ and $q<\frac{4s}{N}$, we investigate the nonexistence of a strong limit for a non-radial solution and $L^2$ concentration phenomenon for a radial solution in Section 5 and Section 6.

2. Preliminaries

Next, we recall some preliminary results that will be used later. Firstly, let us recall the local theory for the problem (1). Note that the parameter $(a,b)$ is fractional admissible if

$$a\in [2,\infty ),b\in [2,\infty ),(a,b)\ne \left(2,\frac{4N-2}{2N-3}\right),\frac{2s}{a}+\frac{N}{b}=\frac{N}{2}.$$

Using Strichartz estimates and the contraction mapping argument, we can prove the well-posedness for (1) in the energy space $H^s$, as in [28].

Proposition 1

(Radial $H^s$ LWP). Let $N\ge 2$, $\frac{N}{2N-1}<s<1$, $b>-2s$, $0<p<\frac{4s-2b}{N-2s}$ and $0<q<\frac{4s}{N-2s}$. Let

$$a_1=\frac{4s(p+2)}{p(N-2s)},b_1=\frac{N(p+2)}{N+ps},$$

$$a_2=\frac{4s(q+2)}{q(N-2s)},b_2=\frac{N(q+2)}{N+qs}.$$

Then, for any $u_0\in H^s$ radial, there exist $T\in (0,+\infty ]$ and a unique local solution to (1) satisfying

$$C([0,T),H^s)\bigcap _{j=1,2}{L}^{a_j}\left([0,T),W^{s,b_j}\right).$$

Moreover, the following properties hold:

• $u\in L_{loc}^a\left([0,T),W^{s,b}\right)$ for any fractional admissible pair $(a,b)$.
• If $T<+\infty$, then ${\parallel u(t)\parallel }_{{\dot{H}}^s}\to +\infty$ as $t\to T$.
• The solution enjoys conservation of mass and energy, i.e., for all $t\in [0,T)$,

  $$\begin{array}{cc}\hfill Mass& :M(u(t)):={\parallel u(t)\parallel }^2=M(u_0);\hfill \\ \hfill Energy& :E(u(t)):=\frac{1}{2}{\parallel u(t)\parallel }_{{\dot{H}}^s}^2-\frac{{\lambda }_1}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t)|}^{p+2}dx\hfill \\ \hfill & -\frac{{\lambda }_2}{q+2}{\parallel u(t)\parallel }_{L^{q+2}}^{q+2}=E(u_0).\hfill \end{array}$$

In fact, the reader can find the proof of this result in [28]. In this case, the Strichartz estimates have no loss of derivatives. We thus obtain a better local well-posedness result compared with the one in the following proposition.

Proposition 2

(Non-radial $H^s$ LWP). Let $s\in (0,1)\setminus \{\frac{1}{2}\}$, $0<p<\frac{4s-2b}{N-2s}$ and $0<q<\frac{4s}{N-2s}$ be such that

$$s>\left\{\begin{array}{c}\frac{1}{2}-\frac{2s}{max(p,q,4)}ifN=1,\hfill \\ \frac{N}{2}-\frac{2s}{max(p,q,2)}ifN\ge 2.\hfill \end{array}\right.$$

Then, for any $u_0\in H^s$, there exist $T\in (0,+\infty ]$ and a unique local solution to (1) satisfying

$$C([0,T),H^s)\bigcap L_{loc}^a\left([0,T),L^{\infty }\right),$$

for some $a>max(p,q,4)$ when $N=1$ and some $a>max(p,q,2)$ when $N\ge 2$. Moreover, the following properties hold:

• If $T<+\infty$, then ${\parallel u(t)\parallel }_{{\dot{H}}^s}\to +\infty$ as $t\to T$.
• The solution enjoys conservation of mass and energy, i.e., $M(u(t))=M(u_0)$ and $E(u(t))=E(u_0)$ for all $t\in [0,T)$.

We refer the reader to [9] (or [10]) for the proof of this result. Note that in the case of non-radial $H^s$ initial data, Strichartz estimates have a loss of derivatives. However, the loss of derivatives can be compensated by using the Sobolev embedding.

Secondly, we collect some classical results needed for this manuscript. We start with a sharp Gagliardo–Nirenberg-type inequality established in [7,16,22].

Lemma 1.

Let $0<p<\frac{4s-2b}{N-2s}$ and $b<min\{2s,N\}$, the Gagliardo–Nirenberg inequality

$${\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u|}^{p+2}dx\le {C(N,p,b,s)\parallel u\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}{\parallel u\parallel }_{L^2}^{(p+2)-\frac{Np+2b}{2s}}$$

holds, and the constant $C(N,p,b,s)$ is explicitly provided by

$$\begin{array}{c}C(N,p,b,s)={\left(\frac{Np+2b}{2s(p+2)-(Np+2b)}\right)}^{\frac{4s-(Np+2b)}{4s}}\frac{2s(p+2)}{{(Np+2b)\parallel Q\parallel }_{L^2}^p},\end{array}$$

(3)

where Q is the nonnegative, nontrivial solution of the equation

$$\begin{array}{c}-{(-\Delta )}^{s}Q-Q+{|x|}^{-b}{|Q|}^{p}Q=0.\end{array}$$

(4)

Moreover, the solution Q satisfies the following relations

$$\begin{array}{c}{\parallel Q\parallel }_{{\dot{H}}^s}={\left(\frac{Np+2b}{2s(p+2)-(Np+2b)}\right)}^{\frac{1}{2}}{\parallel Q\parallel }_{L^2}\end{array}$$

(5)

and

$$\begin{array}{c}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|Q|}^{p+2}dx=\frac{2s(p+2)}{2s(p+2)-(Np+2b)}{\parallel Q\parallel }_{L^2}^2.\end{array}$$

(6)

Remark 1.

When $p=\frac{4s}{N},b=0$, we have

$${\parallel u\parallel }_{L^{\frac{4s+2N}{N}}}^{\frac{4s+2N}{N}}\le {C(N,s)\parallel u\parallel }_{{\dot{H}}^s}^2{\parallel u\parallel }_{L^2}^{\frac{4s}{N}}.$$

and the constant $C(N,s)$ is explicitly provided by

$$\begin{array}{c}C(N,s)=\frac{2s+N}{{N\parallel R\parallel }_{L^2}^{\frac{4s}{N}}},\end{array}$$

(7)

where R is the nonnegative, radially symmetric, decreasing solution of the equation

$$\begin{array}{c}-{(-\Delta )}^{s}R-R+{|R|}^{\frac{4s}{N}}R=0.\end{array}$$

(8)

Moreover, the solution R satisfies the following relations

$$\begin{array}{c}{\parallel R\parallel }_{{\dot{H}}^s}={\left(\frac{N}{2s}\right)}^{\frac{1}{2}}{\parallel R\parallel }_{L^2}\end{array}$$

(9)

and

$$\begin{array}{c}{\parallel R\parallel }_{L^{\frac{4s+2N}{N}}}^{\frac{4s+2N}{N}}=\frac{2s+N}{2s}{\parallel R\parallel }_{L^2}^2.\end{array}$$

(10)

Remark 2.

In fact, in this paper, we also use various other special cases of the Gagliardo–Nirenberg inequality, such as when $b=0$ and when $p=\frac{4s-2b}{N}$. The details will not be described here.

Finally, we provide the profile decomposition of the bounded sequences in $H^s$.

Lemma 2

(Profile Decomposition). Let $N\ge 2$ and $0<s<1$. If ${\{v_n\}}_{n=1}^{+\infty }$ is a bounded sequence in $H^s$, then there exists a subsequence of ${\{v_n\}}_{n=1}^{+\infty }$ (also denoted by ${\{v_n\}}_{n=1}^{+\infty }$ ), a family ${\{x_n^j\}}_{j=1}^{+\infty }$ of sequence ${\mathbb{R}}^N$ and a family ${\left\{V^j\right\}}_{j=1}^{+\infty }$ of $H^s$ functions satisfying the following:

(i)

For every $k\ne j$,

$$\begin{array}{c}|x_n^k-x_n^j|\to +\infty asn\to +\infty .\end{array}$$

(11)

(ii)

For every $l\ge 1$ and every $x\in {\mathbb{R}}^N$, $v_n(x)$ can be decomposed as

$$\begin{array}{c}{v}_n(x)={\displaystyle \sum _{j=1}^l}{V}^j(x-x_n^j)+v_n^l(x)\end{array}$$

(12)

with

$$\begin{array}{c}\underset{l\to +\infty }{lim}\underset{n\to +\infty }{lim\; sup}{\parallel v_n^l\parallel }_{L^p}=0forp\in (2,\frac{2N}{N-2s}).\end{array}$$

(13)

Moreover, as $n\to +\infty$, the following relations hold,

$$\begin{array}{c}{\parallel v_n\parallel }_{L^2}^2={\displaystyle \sum _{j=1}^l}{\parallel V^j\parallel }_{L^2}^2+{\parallel v_n^l\parallel }_{L^2}^2+\mathfrak{o}(1),\\ {\parallel v_n\parallel }_{H^s}^2={\displaystyle \sum _{j=1}^l}{\parallel V^j\parallel }_{H^s}^2+{\parallel v_n^l\parallel }_{H^s}^2+\mathfrak{o}(1),\end{array}$$

(14)

where $\mathfrak{o}(1)$ is such that $\underset{n\to +\infty }{lim}\mathfrak{o}(1)=0$.

Remark 3.

The profile decomposition arguments were proposed by Gérard in [33], Hmidi and Keraani in [30], and Giampiero and Adriano in [34]. The proof of Lemma 2.6 is similar with Proposition 3.1 in [30].

Note that a spatial sequence ${\{x_n^j\}}_{n=1}^{+\infty }\subset {\mathbb{R}}^N$ is called orthogonal if and only if (12) is true for every $j\ne k$.

3. Non-Radial Solution for ${\mathit{\lambda }}_{\mathbf{1}}=\pm \mathbf{1},{\mathit{\lambda }}_{\mathbf{2}}=\mathbf{1}$, $\mathit{p}<\frac{\mathbf{4}\mathit{s}-\mathbf{2}\mathit{b}}{\mathit{N}}$ and $\mathit{q}=\frac{\mathbf{4}\mathit{s}}{\mathit{N}}$

In this section, our purpose is to show the nonexistence of a strong limit at the blow-up time for the problem

$$\left\{\begin{array}{c}i{\partial }_{t}u-{(-\Delta )}^{s}u+{\lambda }_1{|x|}^{-b}{|u|}^{p}u+{|u|}^{\frac{4s}{N}}u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,\hfill \\ u(0,x)=u_0(x),\hfill \end{array}\right.$$

(15)

where ${\lambda }_1=\pm 1,p<\frac{4s-2b}{N}$ and $u_0$ is non-radial. We have the following result.

Theorem 1.

Assume that $u_0\in H^s$ and the solution $u\in C([0,T),H^s)$ of (15) blows up at time T. Then, $u(t)$ does not have a strong limit in $L^2$ as $t\to T$. In addition, we have the stronger property that there is no sequence ${\{t_n\}}_{n=1}^{+\infty }$, such that $t_n\to T$ and $u(t_n)$ converges in $L^2$ as $t_n\to T$.

We remark that we only have to show the following proposition, which is a stronger result than the nonexistence of a limit in $L^2$ at the blow-up time.

Proposition 3.

Let $u(t)$ be the solution of (15) in $C([0,T),H^s)$. Suppose that there is a sequence ${\{t_n\}}_{n=1}^{+\infty }$ such that $t_n\to T$ and $u(t_n)$ has a strong limit in $L^2$ as $n\to +\infty$. Then, ${\parallel u(t)\parallel }_{{\dot{H}}^s}$ belongs to $L^{\infty }(0,T)$.

Proof. 

The proof of Propositon 3 contains two parts.

Part one: In such a case, ${\lambda }_1=1$. Then, due to the conservation of energy, the Gagliardo–Nirenberg inequality and the Sobolev inequality, we have

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2=\frac{2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_n){|}^{p+2}dx+\frac{N}{N+2s}{\parallel u(t_n)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0)\hfill \\ \hfill & \le \frac{2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_n)-u(t_m){|}^{p+2}dx+\frac{2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_m)|}^{p+2}dx\hfill \\ \hfill & +\frac{N}{N+2s}\parallel u(t_n)-u(t_m){\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+\frac{N}{N+2s}{\parallel u(t_m)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0)\hfill \\ \hfill & \le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{p+2-\frac{Np+2b}{2s}}\left(\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}+{\parallel u(t_m)\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}\right)\hfill \\ \hfill & +\frac{2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_m){|}^{p+2}dx+C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s}{N}}(\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\hfill \\ \hfill & +\parallel u(t_m){\parallel }_{{\dot{H}}^s}^2)+\frac{N}{N+2s}{\parallel u(t_m)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0).\hfill \end{array}$$

Hence, some computation and the conservation law yield for a fixed m

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{p+2-\frac{Np+2b}{2s}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}\hfill \\ \hfill & +C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s}{N}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_m\hfill \end{array}$$

(16)

for every n, where $C_m$ depends on m and the space dimension N, s, p, and b.

Now, let us show that there is a $C>0$, such that $\parallel u(t_n){\parallel }_{{\dot{H}}^s}\le C$ for every n. As the sequence $u(t_n)$ converges strongly in $L^2$, there is a positive $n_0$, such that for all $n\ge n_0$, $m\ge n_0$,

$$C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{p+2-\frac{Np+2b}{2s}}<\frac{1}{2},C{\parallel u(t_n)-u(t_m)\parallel }_{L^2}^{\frac{4s}{N}}<\frac{1}{2},$$

where we use the condition $p+2-\frac{Np+2b}{2s}>0$. Therefore, choosing $m=n_0$ in (16), we obtain

$$\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\le \frac{1}{2}\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}+\frac{1}{2}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_{n_0},$$

which implies with the conservation law and $\frac{Np+2b}{2s}<2$ that the sequence $u(t_n)$ is bounded in $H^s$.

Part two: In such a case, ${\lambda }_1=-1$. Then, in a similar way, we have

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2=\frac{-2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_n){|}^{p+2}dx+\frac{N}{N+2s}{\parallel u(t_n)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0)\hfill \\ \hfill & \le \frac{N}{N+2s}\parallel u(t_n)-u(t_m){\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+\frac{N}{N+2s}{\parallel u(t_m)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0)\hfill \\ \hfill & \le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s}{N}}(\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\hfill \\ \hfill & +\parallel u(t_m){\parallel }_{{\dot{H}}^s}^2)+\frac{N}{N+2s}{\parallel u(t_m)\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}+2E(u_0).\hfill \end{array}$$

Hence, some computation and the conservation law yield for a fixed m

$$\begin{array}{cc}\hfill & {\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2\le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s}{N}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_m\hfill \end{array}$$

for every n, where $C_m$ depends on m and the space dimension N, s, p, and b.

Now, let us show that there is a $C>0$, such that ${\parallel u(t_n)\parallel }_{{\dot{H}}^s}\le C$ for every n. As the sequence $u(t_n)$ converges strongly in $L^2$, there is a positive $n_0$, such that for all $n\ge n_0$, $m\ge n_0$,

$$C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s}{N}}<\frac{1}{2}.$$

Therefore, choosing $m=n_0$, we obtain

$${\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2\le \frac{1}{2}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_{n_0},$$

which implies with the conservation law that the sequence $u(t_n)$ is bounded in $H^s$.

Finally, the unique local solution in $H^s$ for problem (15) and the boundedness of $u(t_n)$ in $H^s$ yield Proposition (3). In fact, we note that the existence time of the local solution of problem (15) depends only on the size of the $H^s$ norm of the initial data. □

4. Radial Solution for ${\mathit{\lambda }}_{\mathbf{1}}=\pm \mathbf{1},{\mathit{\lambda }}_{\mathbf{2}}=\mathbf{1}$, $\mathit{p}<\frac{\mathbf{4}\mathit{s}-\mathbf{2}\mathit{b}}{\mathit{N}}$ and $\mathit{q}=\frac{\mathbf{4}\mathit{s}}{\mathit{N}}$

In this section, our purpose is to show the $L^2$ concentration phenomenon for the problem (15) if $u_0\in H^s$ is radial. In this case, we will obtain a stronger conclusion than Theorem (1).

Theorem 2.

Let $N\ge 2$, $s\in (\frac{1}{2},1)$. If the solution of (15) blows up in finite time $T>0$. Let $a(t)$ be a real-valued nonnegative function defined on $[0,T)$, satisfying ${a(t)\parallel u(t)\parallel }_{{\dot{H}}^s}^{\frac{1}{s}}\to \infty$ as $t\to T$. Then there exists $x(t)\in {\mathbb{R}}^N$ such that

$$\begin{array}{c}\underset{t\to T}{lim\; inf}{\int }_{|x-x(t)|\le a(t)}{|u(t,x)|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{|R|}^{2}dx,\end{array}$$

(17)

where R is the ground state solution of (8).

To prove (2), we first prove the following compactness lemma. It implies that if a bounded sequence in some strong norm (e.g., $H^s$) does not converge weakly to zero in a weaker norm, then this can be attributed to the sequence containing a profile of concentration. Although this lemma has been proven in many literature works (see [29]), we provide the main steps of its proof for the sake of completeness.

Lemma 3

(Compactness Lemma). Assume $N\ge 2$ and $\frac{N}{2N-1}<s\le 1$. Let ${\{u_n\}}_{n=1}^{+\infty }$ be a bounded family of $H^s$ such that

$$\underset{n\to +\infty }{lim\; sup}{\parallel u_n\parallel }_{{\dot{H}}^s}\le M,\underset{n\to +\infty }{lim\; sup}{\parallel u_n\parallel }_{L^{\frac{2N+4s}{N}}}\ge m>0.$$

Then, up to a subsequence,

$$u_n(·)\rightharpoonup Vweaklyin{H}^s,$$

for some $V\in H^s$ satisfying

$${\parallel V\parallel }_{L^2}\ge {(\frac{N}{(N+2s)M^2})}^{\frac{N}{4s}}{m}^{\frac{N+2s}{2s}}{\parallel R\parallel }_{L^2},$$

where R is the ground state solution of (8).

Proof. 

Due to the profile decomposition (12) and the sharp Gagliardo–Nirenberg inequality, we have

$$\begin{array}{cc}\hfill & m^{\frac{2N+4s}{N}}=\underset{n\to +\infty }{lim\; sup}{\parallel u_n\parallel }_{L^{\frac{2N+4s}{N}}}^{\frac{2N+4s}{N}}\hfill \\ \hfill & \le \parallel {\mathrm{\Sigma }}_{j=1}^{+\infty }{V}^j(x-x_n^j){\parallel }_{L^{\frac{2N+4s}{N}}}^{\frac{2N+4s}{N}}\le {\mathrm{\Sigma }}_{j=1}^{+\infty }{\parallel V^j\parallel }_{L^{\frac{2N+4s}{N}}}^{\frac{2N+4s}{N}}\hfill \\ \hfill & \le {\mathrm{\Sigma }}_{j=1}^{+\infty }\frac{2s+N}{{N\parallel R\parallel }_{L^2}^{\frac{4s}{N}}}\parallel V^j{\parallel }_{L^2}^{\frac{4s}{N}}{\parallel V^j\parallel }_{{\dot{H}}^s}^2\hfill \\ \hfill & \le \frac{2s+N}{{N\parallel R\parallel }_{L^2}^{\frac{4s}{N}}}\underset{j\ge 1}{sup}\{\parallel V^j{\parallel }_{L^2}^{\frac{4s}{N}}\}{\mathrm{\Sigma }}_{j=1}^{+\infty }{\parallel V^j\parallel }_{{\dot{H}}^s}^2.\hfill \end{array}$$

(18)

Applying the profile decomposition again, we obtain

$$\begin{array}{c}{\mathrm{\Sigma }}_{j=1}^{+\infty }{\parallel V^j\parallel }_{{\dot{H}}^s}^2\le \underset{n\to +\infty }{lim\; sup}{\parallel u_n\parallel }_{{\dot{H}}^s}^2\le M^2.\end{array}$$

(19)

Hence, it follows from (18) and (19) that

$$\underset{j\ge 1}{sup}\{\parallel V^j{\parallel }_{L^2}^{\frac{4s}{N}}\}\ge \frac{m^{\frac{4s}{N}+2}N{\parallel R\parallel }_{L^2}^{\frac{4s}{N}}}{(2s+N)M^2}.$$

By the convergence of series ${\mathrm{\Sigma }}_{j=1}^{+\infty }{\parallel V^j\parallel }_{L^2}^2$, there exists $j_0\ge 1$, such that

$${\parallel V^{j_0}\parallel }_{L^2}^{\frac{4s}{N}}=\underset{j\ge 1}{sup}\{\parallel V^j{\parallel }_{L^2}^{\frac{4s}{N}}\}$$

and

$$\parallel V^{j_0}{\parallel }_{L^2}^{\frac{4s}{N}}\ge \frac{m^{\frac{4s}{N}+2}N{\parallel R\parallel }_{L^2}^{\frac{4s}{N}}}{(2s+N)M^2}.$$

Finally, a change of spatial translation and profile decomposition provides

$$u_n(x+x_n^{j_0})=V^{j_0}(x)+\sum _{j\ne j_0}{V}^j(x+x_n^{j_0}-x_n^j)+v_n^l(x+x_n^{j_0}).$$

Using the pairwise orthogonality of ${\{x_n^j\}}_{j=1}^{+\infty }$, we have $V^j(·+x_n^{j_0}-x_n^j)$, which converges weakly to 0 for every $j\ne j_0$ in $H^s$. And $u_n(·+x_n^{j_0})$ converges weakly to $V^{j_0}+{\tilde{v}}^l$ in $H^s$, where ${\tilde{v}}^l$ is the weak limit of $v_n^l(x+x_n^{j_0})$. However,

$${\parallel {\tilde{v}}^l\parallel }_{L^p}\le \underset{n\to +\infty }{lim\; sup}{\parallel v_n^l\parallel }_{L^p}\to 0.$$

Following the uniqueness of the weak limit, we obtain ${\tilde{v}}^l=0$ for all $l\ge j_0$ and $u_n(·+x_n^{j_0})$ converges weakly to $V^{j_0}$ in $H^s$. This completes the proof. □

Remark 4.

The lower bound on the $L^2$-norm of V is optimal. In fact, if we take $v_n=Q$, then we obtain equality.

By applying the compactness lemma, we can prove Theorem 2.

Proof. 

In the sequel, we use the following notation:

$$\begin{array}{c}\hfill {\rho }^s(t)=\frac{{\parallel R\parallel }_{{\dot{H}}^s}}{{\parallel u(t)\parallel }_{{\dot{H}}^s}},v(x,t)={\rho }^{\frac{N}{2}}(t)u(t,\rho x).\end{array}$$

(20)

Let ${\{t_n\}}_{j=1}^{+\infty }$ be an any time sequence such that $t_n\to T$, ${\rho }_n=\rho (t_n)$ and $v_n=v(t_n)$. Then, the family ${\{v_n\}}_{n=1}^{+\infty }$ satisfies

$$\begin{array}{c}\hfill \parallel v_n{\parallel }_{L^2}=\parallel u(t_n){\parallel }_{L^2}=\parallel u_0{\parallel }_{L^2},\parallel v_n{\parallel }_{{\dot{H}}^s}={\rho }_n^s\parallel u(t_n){\parallel }_{{\dot{H}}^s}={\parallel Q\parallel }_{{\dot{H}}^s}.\end{array}$$

(21)

Observe that

$$\begin{array}{cc}\hfill & H(v_n):=\frac{1}{2}\parallel v_n{\parallel }_{{\dot{H}}^s}^2-\frac{N}{4s+2N}{\parallel v_n\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}\hfill \\ \hfill & ={\rho }_n^{2s}\left(\frac{1}{2}\parallel u_n{\parallel }_{{\dot{H}}^s}^2-\frac{N}{4s+2N}{\parallel u_n\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}\right)\hfill \\ \hfill & ={\rho }_n^{2s}\left(E(u_0)+\frac{{\lambda }_1}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_n,x)|}^{p+2}dx\right).\hfill \end{array}$$

(22)

Furthermore, by the Gagliardo–Nirenberg inequality (Lemma 2.2), we have

$$\begin{array}{cc}\hfill & |H(v_n)|\le {\rho }_n^{2s}|E(u_0)|+{\rho }_n^{2s}\frac{1}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_n,x)|}^{p+2}dx\hfill \\ \hfill & \le {\rho }_n^{2s}|E(u_0)|+{\rho }_n^{2s}C(N,p,b,s)\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}}{\parallel u(t_n)\parallel }_{L^2}^{(p+2)-\frac{Np+2b}{2s}}\hfill \\ \hfill & ={\rho }_n^{2s}|E(u_0)|+C(N,p,b,s){\parallel R\parallel }_{{\dot{H}}^s}^2\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Np+2b}{2s}-2}{\parallel u_0\parallel }_{L^2}^{(p+2)-\frac{Np+2b}{2s}}.\hfill \end{array}$$

(23)

Note that $\underset{n\to +\infty }{lim}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}=\infty ,$$\underset{n\to +\infty }{lim}{\rho }_n^{2s}=0$ and $\frac{Np+2b}{2s}-2<0$. Hence, we obtain

$$\underset{n\to +\infty }{lim}H(v_n)=0,$$

which implies $\underset{n\to +\infty }{lim}\parallel v_n{\parallel }_{L^{\frac{4s}{N}+2}}^{\frac{4s}{N}+2}=\frac{2s+N}{N}{\parallel R\parallel }_{{\dot{H}}^s}^2$. Now, set $M={\parallel R\parallel }_{{\dot{H}}^s}$ and $m=$${\left(\frac{2s+N}{N}{\parallel R\parallel }_{{\dot{H}}^s}^2\right)}^{\frac{N}{4s+2N}}$, we can apply the compactness lemma for the sequence $\{v(t_n)\}$ in $H^s$. And we obtain $V\in H^s$ and ${\{x_n\}}_{n=1}^{+\infty }$ such that, up to a subsequence, $v_n(·+x_n)={\rho }_n^{\frac{N}{2}}u(t_n,{\rho }_n(·+x_n))$ converges weakly to V in $H^s$. The weak limit V satisfies ${\parallel V\parallel }_{L^2}\ge {\parallel R\parallel }_{L^2}$. Due to ${a(t)\parallel u(t)\parallel }_{{\dot{H}}^s}^{\frac{1}{s}}\to \infty$ as $t\to T$, we have

$$\frac{a(t_n)}{{\rho }_n}=\frac{a(t_n){\parallel u(t)\parallel }_{{\dot{H}}^s}^{\frac{1}{s}}}{{\parallel R\parallel }_{{\dot{H}}^s}^{\frac{1}{s}}}\to \infty \mathrm{a}\mathrm{s}n\to +\infty .$$

Hence, for every $r>0$, there exists $n_0>0$ such that for every $n>n_0$, $r{\rho }_n<a(t_n)$. So, we have

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a(t_n)}|u(t_n,x){|}^{2}dx\ge \underset{n\to \infty }{lim\; inf}{\int }_{|x-x_n|\le r{\rho }_n}{|u(t_n,x)|}^{2}dx\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}{\int }_{|x|\le r}{\rho }_n^N{|u(t_n,{\rho }_n(x+x_n))|}^{2}dx\hfill \\ \hfill & \ge \underset{n\to \infty }{lim\; inf}{\int }_{|x|\le r}{|V(x)|}^{2}dx\hfill \\ \hfill & \ge {\parallel V\parallel }_{L^2}^2,\hfill \end{array}$$

where we use the arbitrariness of $r>0$. And, as $\{t_n\}$ is arbitrary, we can obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a(t)}{|u(t,x)|}^{2}dx\ge {\parallel V\parallel }_{L^2}^2\ge {\parallel R\parallel }_{L^2}^2.\end{array}$$

(24)

Using the continuity of this function $g(y)={\int }_{|x-y|\le a(t)}{|u(t,x)|}^{2}dx$ for every $t\in [0,T)$ and the fact $\underset{|y|\to \infty }{lim}g(y)=0$. There exists a function $x(t)\in {\mathbb{R}}^N$ such that for every $t\in [0,T)$

$$\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{|x-y|\le a(t)}{|u(t,x)|}^{2}dx={\int }_{|x-x(t)|\le a(t)}{|u(t,x)|}^{2}dx.$$

This and (24) yield (17). The proof is completed. □

Remark 5.

Indeed, we can choose $a(t)=\frac{1}{{\parallel u(t)\parallel }_{{\dot{H}}^s}^{\frac{1}{s}-ϵ}}$ with $0<ϵ<\frac{1}{s}$. It is obvious that $a(t)$ satisfies the condition in Theorem 2.

Next, using the uniqueness of radial solution of problem (8), we study the limiting profile of the radial blow-up solutions of problem (15).

Theorem 3.

Let $N\ge 2$, $s\in (\frac{1}{2},1)$, $\parallel u_0{\parallel }_{L^2}={\parallel R\parallel }_{L^2}$. If the solution of (15) blows up in finite time $T>0$. Then, there exists $x(t)\in {\mathbb{R}}^N$ and $\theta (t)\in [0,2\pi )$ such that

$${\rho }^{\frac{N}{2}}(t)u(t,\rho (t)(·+x(t)))e^{i\theta (t)}\to Rstronglyin{H}^s,ast\to T,$$

where R is the ground state solution of (8) and $\rho (t)=\frac{{\parallel R\parallel }_{{\dot{H}}^s}}{{\parallel u(t)\parallel }_{{\dot{H}}^s}}$.

As the proof of the Theorem 3 is similar to the one in [29,30], we omit it.

Remark 6.

We should note that when $p<\frac{4s-2b}{N}$, there is a gap between the upper bound of p and the $L^2$-critical index $\frac{4s}{N}$. Unfortunately, we do not have limit profile results when $\frac{4s-2b}{N}\le p<\frac{4s}{N}$. This situation does not occur in the general fractional Schrödinger equation with a combined non-linearity, i.e., $b=0$.

5. Non-Radial Solution for ${\mathit{\lambda }}_{\mathbf{1}}=\mathbf{1},{\mathit{\lambda }}_{\mathbf{2}}=\pm \mathbf{1}$, $\mathit{p}=\frac{\mathbf{4}\mathit{s}-\mathbf{2}\mathit{b}}{\mathit{N}}$ and $\mathit{q}<\frac{\mathbf{4}\mathit{s}}{\mathit{N}}$

In this section, our purpose is to show the nonexistence of a strong limit at the blow-up time for the problem

$$\left\{\begin{array}{c}i{\partial }_{t}u-{(-\Delta )}^{s}u+{|x|}^{-b}{|u|}^{\frac{4s-2b}{N}}u+{\lambda }_2{|u|}^{q}u=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,\hfill \\ u(0,x)=u_0(x),\hfill \end{array}\right.$$

(25)

where ${\lambda }_2=\pm 1,q<\frac{4s}{N}$ and $u_0$ is non-radial. Unlike problem (15), the $L^2$-critical term of this problem appears in the inhomogeneous term, which brings to the problem. But we also have the following result.

Theorem 4.

Assume that $u_0\in H^s$ and the solution $u\in C([0,T),H^s)$ of (25) blows up at time T. Then, $u(t)$ does not have a strong limit in $L^2$ as $t\to T$. In addition, we have the stronger property that there is no sequence ${\{t_n\}}_{n=1}^{+\infty }$, such that $t_n\to T$ and $u(t_n)$ converges in $L^2$ as $t_n\to T$.

Similar to the proof of Theorem 1, we only have to show the following proposition, which is a stronger result than the nonexistence of a limit in $L^2$ at the blow-up time.

Proposition 4.

Let $u(t)$ be the solution of (25) in $C([0,T),H^s)$. Suppose that there is a sequence ${\{t_n\}}_{n=1}^{+\infty }$, such that $t_n\to T$ and $u(t_n)$ has a strong limit in $L^2$ as $n\to +\infty$. Then, ${\parallel u(t)\parallel }_{{\dot{H}}^s}$ belongs to $L^{\infty }(0,T)$.

Proof. 

The proof of Propositon 4 contains two parts.

Part one: In such a case, ${\lambda }_2=1$. Then, due to conservation of energy, the Gagliardo–Nirenberg inequality and the Sobolev inequality, we have

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2=\frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_n){|}^{\frac{4s-2b+2N}{N}}dx+\frac{2}{q+2}{\parallel u(t_n)\parallel }_{L^{q+2}}^{q+2}+2E(u_0)\hfill \\ \hfill & \le \frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_n)-u(t_m)|}^{\frac{4s-2b+2N}{N}}dx+\hfill \\ \hfill & \frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_m){|}^{\frac{4s-2b+2N}{N}}dx+\frac{2}{q+2}{\parallel u(t_n)-u(t_m)\parallel }_{L^{q+2}}^{q+2}\hfill \\ \hfill & +\frac{2}{q+2}{\parallel u(t_m)\parallel }_{L^{q+2}}^{q+2}+2E(u_0)\hfill \\ \hfill & \le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s-2b}{N}}\left(\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2+{\parallel u(t_m)\parallel }_{{\dot{H}}^s}^2\right)\hfill \\ \hfill & +\frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_m){|}^{\frac{4s-2b+2N}{N}}dx+C{\parallel u(t_n)-u(t_m)\parallel }_{L^2}^{q+2-\frac{Nq}{2s}}\hfill \\ \hfill & \times (\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}}+\parallel u(t_m){\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}})+\frac{2}{q+2}{\parallel u(t_m)\parallel }_{L^{q+2}}^{q+2}+2E(u_0).\hfill \end{array}$$

Hence, some computation and the conservation law yield for a fixed m

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s-2b}{N}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2\hfill \\ \hfill & +C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{q+2-\frac{Nq}{2s}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}}+C_m\hfill \end{array}$$

(26)

for every n, where $C_m$ depends on m and the space dimension N, s, p, and b.

Now, let us show that there is a $C>0$, such that $\parallel u(t_n){\parallel }_{{\dot{H}}^s}\le C$ for every n. As the sequence $u(t_n)$ converges strongly in $L^2$, there is a positive $n_0$, such that for all $n\ge n_0$, $m\ge n_0$,

$$C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{q+2-\frac{Nq}{2s}}<\frac{1}{2},C{\parallel u(t_n)-u(t_m)\parallel }_{L^2}^{\frac{4s-2b}{N}}<\frac{1}{2},$$

where we use the condition $q+2-\frac{Nq}{2s}>0$. Therefore, choosing $m=n_0$ in (26), we obtain

$$\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\le \frac{1}{2}\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2+\frac{1}{2}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}}+C_{n_0},$$

which implies with the conservation law and $\frac{Nq}{2s}<2$ that the sequence $u(t_n)$ is bounded in $H^s$.

Part two: In such a case, ${\lambda }_2=-1$. Then, by a similar way, we have

$$\begin{array}{cc}\hfill & \parallel u(t_n){\parallel }_{{\dot{H}}^s}^2=\frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u(t_n){|}^{\frac{4s-2b+2N}{N}}dx-\frac{2}{q+2}{\parallel u(t_n)\parallel }_{L^{q+2}}^{q+2}+2E(u_0)\hfill \\ \hfill & \le \le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s-2b}{N}}\left(\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2+{\parallel u(t_m)\parallel }_{{\dot{H}}^s}^2\right)\hfill \\ \hfill & +\frac{N}{2s-b+N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_m)|}^{\frac{4s-2b+2N}{N}}dx+2E(u_0).\hfill \end{array}$$

Hence, some computation and the conservation law yield for a fixed m

$$\begin{array}{cc}\hfill & {\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2\le C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s-2b}{N}}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_m\hfill \end{array}$$

for every n, where $C_m$ depends on m and the space dimensions N, s, p, and b.

Now, let us show that there is a $C>0$, such that ${\parallel u(t_n)\parallel }_{{\dot{H}}^s}\le C$ for every n. As the sequence $u(t_n)$ converges strongly in $L^2$, there is a positive $n_0$ such that for all $n\ge n_0$, $m\ge n_0$,

$$C\parallel u(t_n)-u(t_m){\parallel }_{L^2}^{\frac{4s-2b}{N}}<\frac{1}{2}.$$

Therefore, choosing $m=n_0$, we obtain

$$\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2\le \frac{1}{2}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}^2+C_{n_0},$$

which implies with the conservation law that the sequence $u(t_n)$ is bounded in $H^s$. This completes the proof of Proposition (4). □

6. Radial Solution for ${\mathit{\lambda }}_{\mathbf{1}}=\mathbf{1},{\mathit{\lambda }}_{\mathbf{2}}=\pm \mathbf{1}$, $\mathit{p}=\frac{\mathbf{4}\mathit{s}-\mathbf{2}\mathit{b}}{\mathit{N}}$ and $\mathit{q}<\frac{\mathbf{4}\mathit{s}}{\mathit{N}}$

In this section, our purpose is to show $L^2$ concentration phenomenon for the problem (25) if $u_0\in H^s$ is radial. Unlike Theorem 2, due to the presence of inhomogeneous terms, the mass concentrates at the origin when $b>0$. In a sense, we could consider that the origin is a blow-up point for the solution of (25).

Theorem 5.

Let $N\ge 2$, $s\in (\frac{1}{2},1)$. If the solution of (25) blows up in finite time $T>0$. Let $a(t)$ be a real-valued nonnegative function defined on $[0,T)$ satisfying ${a(t)\parallel u(t)\parallel }_{{\dot{H}}^s}^{\frac{1}{s}}\to \infty$ as $t\to T$. Then,

$$\begin{array}{c}\underset{t\to T}{lim\; inf}{\int }_{|x|\le a(t)}{|u(t,x)|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{|Q|}^{2}dx,\end{array}$$

(27)

where Q is the ground state solution of (4) when $p=\frac{4s-2b}{N}$.

To prove (5), we first prove the following compactness lemma. Unlike Lemma 3, we must look for some weighted compactness lemma as the presence of inhomogeneous terms. And to prove this lemma, we have to apply some different methods.

Lemma 4

(Weighted compactness lemma). Assume $N\ge 2$ and $\frac{N}{2N-1}<s\le 1$. Let ${\{u_n\}}_{n=1}^{+\infty }$ be a radial bounded family of $H^s$ such that

$$\underset{n\to +\infty }{lim\; sup}\parallel u_n{\parallel }_{{\dot{H}}^s}\le M,\underset{n\to +\infty }{lim\; sup}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t)|}^{\frac{4s-2b+2N}{N}}dx\ge m>0.$$

Then, up to a subsequence,

$$u_n(·)\rightharpoonup Vweaklyin{H}^s,$$

for some radial function $V\in H^s$ satisfying

$${\parallel V\parallel }_{L^2}\ge {\left(\frac{mN}{(N+2s-b)M^2}\right)}^{\frac{N}{4s-2b}}{\parallel Q\parallel }_{L^2},$$

where Q is the ground state solution of (4) when $p=\frac{4s-2b}{N}$.

Proof. 

Indeed, as ${\{u_n\}}_{n=1}^{+\infty }$ is a bounded radial sequence in $H^s$, there exists $V\in H^s$, such that up to a sequence, $u_n\rightharpoonup V$ weakly in $H^s$. Firstly, we claim that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u_n{|}^{\frac{2N+4s-2b}{N}}dx\to {\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|V|}^{\frac{2N+4s-2b}{N}}dx,\end{array}$$

(28)

as $n\to +\infty$. Indeed, let $ϵ>0$, since ${\{u_n\}}_{n=1}^{+\infty }$ is bounded in $H^s$, we have for any $R>0$,

$$\begin{array}{cc}\hfill & \left|{\int }_{|x|\ge R}{|x|}^{-b}(|u_n{|}^{\frac{2N+4s-2b}{N}}-{|V|}^{\frac{2N+4s-2b}{N}})dx\right|\hfill \\ \hfill & \le R^{-b}(\parallel u_n{\parallel }_{L^{\frac{2N+4s-2b}{N}}}^{\frac{2N+4s-2b}{N}}+{\parallel V\parallel }_{L^{\frac{2N+4s-2b}{N}}}^{\frac{2N+4s-2b}{N}})\hfill \\ \hfill & \le C{R}^{-b}(\parallel u_n{\parallel }_{H^s}^{\frac{2N+4s-2b}{N}}+{\parallel V\parallel }_{H^s}^{\frac{2N+4s-2b}{N}})\hfill \\ \hfill & \le C{R}^{-b},\hfill \end{array}$$

where we use the Sobolev embedding $H^s\hookrightarrow L^{\frac{2N+4s-2b}{N}}$. By choosing $R>0$ sufficiently large, we have

$$\begin{array}{c}\hfill \left|{\int }_{|x|\ge R}{|x|}^{-b}(|u_n{|}^{\frac{2N+4s-2b}{N}}-{|V|}^{\frac{2N+4s-2b}{N}})dx\right|<\frac{ϵ}{2}.\end{array}$$

(29)

On the other hand, we have

$$\begin{array}{cc}\hfill & \left|{\int }_{|x|\le R}{|x|}^{-b}(|u_n{|}^{\frac{2N+4s-2b}{N}}-{|V|}^{\frac{2N+4s-2b}{N}})dx\right|\hfill \\ \hfill & \le {\parallel |x|}^{-b}{\parallel }_{L^{\delta }(|x|\le R)}\parallel |u_n{|}^{\frac{2N+4s-2b}{N}}-{|V|}^{2\sigma +2}{\parallel }_{L^{\mu }(|x|\le R)}\hfill \end{array}$$

provided that $\delta ,\mu \ge 1,1=\frac{1}{\delta }+\frac{1}{\mu }$. The term ${\parallel |x|}^{-b}{\parallel }_{L^{\delta }(|x|\le R)}$ is bounded if $\frac{N}{\delta }>b$. Thus $\frac{1}{\delta }>\frac{b}{N}$ and $\frac{1}{\mu }=1-\frac{1}{\delta }<\frac{N-b}{N}$. Next, we bound

$$\begin{array}{cc}\hfill & \parallel |u_n{|}^{\frac{2N+4s-2b}{N}}-{|V|}^{\frac{2N+4s-2b}{N}}{\parallel }_{L^{\mu }(|x|\le R)}\hfill \\ \hfill & \le C(\parallel u_n{\parallel }_{L^{\rho }}^{\frac{N+4s-2b}{N}}+{\parallel V\parallel }_{L^{\rho }}^{\frac{N+4s-2b}{N}})\parallel u_n{-V\parallel }_{L^{\rho }(|x|\le R)},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \frac{2N+4s-2b}{\rho N}=\frac{1}{\mu }<\frac{N-b}{N}.\end{array}$$

(30)

By the Sobolev embedding $H^s\hookrightarrow L^r$ for any $2\le r<\frac{2N}{N-2s}$ and the fact that $u_n\to v$ strongly in $L^r(|x|\le R)$ for any $2\le r<\frac{2N}{N-2s}$, we are able to choose $\rho \in (2,\frac{2N}{N-2s})$ so that (30) holds. Indeed, we can choose $\rho$, which is small but close to $\frac{2N}{N-2s}$. Furthermore, due to $\frac{(N+2s-b)(N-2s)}{N}<\frac{N-b}{N},$ we see that (30) is satisfied. As a consequence, we obtain

$$\begin{array}{c}\hfill \left|{\int }_{|x|\le R}{|x|}^{-b}(|u_n{|}^{\frac{N+4s-2b}{N}}-{|V|}^{\frac{N+4s-2b}{N}})dx\right|\le C{\parallel u_n-V\parallel }_{L^{\rho }(|x|\le R)}<\frac{ϵ}{2}\end{array}$$

(31)

for n when it is sufficiently large, Collecting (29) and (31), we prove (28).

Therefore, the sharp Gagliardo–Nirenberg inequality yields

$$\begin{array}{cc}\hfill & m\le \underset{n\to +\infty }{lim\; sup}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|u_n{|}^{\frac{2N+4s-2b}{N}}dx={\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|V|}^{\frac{2N+4s-2b}{N}}dx\hfill \\ \hfill & \le \frac{N+2s-b}{N}\frac{1}{{\parallel Q\parallel }_{L^2}^{\frac{4s-2b}{N}}}{\parallel V\parallel }_{{\dot{H}}^s}^2{\parallel V\parallel }_{L^2}^{\frac{4s-2b}{N}}.\hfill \end{array}$$

(32)

As $v_n\rightharpoonup V$ weakly in $H^s$, the semi-continuity of weak convergence implies

$${\parallel V\parallel }_{{\dot{H}}^s}\le \underset{n\to +\infty }{lim\; sup}{\parallel u_n\parallel }_{{\dot{H}}^s}\le M.$$

Hence

$${\parallel V\parallel }_{L^2}^{\frac{4s-2b}{N}}\ge \frac{{mN\parallel Q\parallel }_{L^2}^{\frac{4s-2b}{N}}}{(N+2s-b)M^2}.$$

This completes the proof of Lemma 4. □

Thanks to Lemma 4, next, we provide the proof of the $L^2$-concentration result.

Proof. 

Set

$$\begin{array}{c}\hfill {\rho }^s(t)=\frac{{\parallel Q\parallel }_{{\dot{H}}^s}}{{\parallel u(t)\parallel }_{{\dot{H}}^s}},v(x,t)={\rho }^{\frac{N}{2}}u(\rho x,t).\end{array}$$

(33)

Let ${\{t_n\}}_{j=1}^{+\infty }$ be an any time sequence, such that $t_n\to T$, ${\rho }_n=\rho (t_n)$ and $v_n=v(t_n)$. Therefore, the family ${\{v_n\}}_{n=1}^{+\infty }$ satisfies

$$\begin{array}{c}\hfill \parallel v_n{\parallel }_{L^2}=\parallel u(t_n){\parallel }_{L^2}=\parallel u_0{\parallel }_{L^2},\parallel v_n{\parallel }_{{\dot{H}}^s}={\rho }_n^s\parallel u(t_n){\parallel }_{{\dot{H}}^s}={\parallel Q\parallel }_{{\dot{H}}^s}.\end{array}$$

(34)

Note that

$$\begin{array}{cc}\hfill & H(v_n):=\frac{1}{2}\parallel v_n{\parallel }_{{\dot{H}}^s}^2-\frac{N}{4s-2b+2N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|v_n|}^{\frac{2N+4s-2b}{N}}dx\hfill \\ \hfill & ={\rho }_n^{2s}\left(\frac{1}{2}\parallel u(t_n){\parallel }_{{\dot{H}}^s}^2-\frac{N}{4s-2b+2N}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}{|u(t_n)|}^{\frac{2N+4s-2b}{N}}dx\right)\hfill \\ \hfill & ={\rho }_n^{2s}\left(E(u_0)+\frac{{\lambda }_2}{q+2}{\parallel u(t_n)\parallel }_{L^{q+2}}^{q+2}\right).\hfill \end{array}$$

(35)

Furthermore, from the Gagliardo–Nirenberg inequality (Lemma 2.2), we have

$$\begin{array}{cc}\hfill & |H(v_n)|\le {\rho }_n^{2s}|E(u_0)|+{\rho }_n^{2s}\frac{1}{q+2}{\parallel u_n\parallel }_{L^{q+2}}^{q+2}\hfill \\ \hfill & \le {\rho }_n^{2s}|E(u_0)|+{\rho }_n^{2s}C(N,q,s)\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}}{\parallel u(t_n)\parallel }_{L^2}^{(q+2)-\frac{Nq}{2s}}\hfill \\ \hfill & ={\rho }_n^{2s}|E(u_0)|+C(N,q,s){\parallel R\parallel }_{{\dot{H}}^s}^2\parallel u(t_n){\parallel }_{{\dot{H}}^s}^{\frac{Nq}{2s}-2}{\parallel u_0\parallel }_{L^2}^{(q+2)-\frac{Nq}{2s}}.\hfill \end{array}$$

(36)

As $\underset{n\to +\infty }{lim}{\parallel u(t_n)\parallel }_{{\dot{H}}^s}=\infty ,$$\underset{n\to +\infty }{lim}{\rho }_n^{2s}=0$ and $\frac{Nq}{2s}-2<0$. We obtain

$$\underset{n\to +\infty }{lim}H(v_n)=0,$$

which implies

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}{\int }_{{\mathbb{R}}^N}{|x|}^{-b}|v_n{|}^{\frac{2N+4s-2b}{N}}dx=\frac{2s-b+N}{N}{\parallel Q\parallel }_{{\dot{H}}^s}^2.\end{array}$$

(37)

Set $m=\frac{N+2s-b}{N}{\parallel Q\parallel }_{{\dot{H}}^s}^2$ and $M={\parallel Q\parallel }_{{\dot{H}}^s}$. Then, it follows from the compactness lemma that there exist V and ${\{{\tilde{t}}_n\}}_{n=1}^{+\infty }\subset \mathbb{R}$ such that, up to a subsequence,

$$\begin{array}{c}\hfill v_n(x,t)={\rho }_n^{\frac{N}{2}}u({\rho }_{n}x,t)\rightharpoonup V\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{l}\mathrm{y}\mathrm{in}{H}^s\end{array}$$

(38)

with

$$\begin{array}{c}\hfill {\parallel V\parallel }_{L^2}\ge {\parallel Q\parallel }_{L^2}.\end{array}$$

(39)

Observe that

$$\begin{array}{c}\hfill \frac{a(t_n)}{{\rho }_n}=\frac{a(t_n){\parallel u(t_n)\parallel }_{{\dot{H}}^s}}{{\parallel Q\parallel }_{{\dot{H}}^s}}\mathrm{a}\mathrm{s},n\to +\infty .\end{array}$$

(40)

Then for every $r>0$, there exists $n_0>0$ such that for every $n>n_0$, $r{\rho }_n<a(t_n)$. Hence, using (38), we get

$$\begin{array}{cc}\hfill & \underset{n\to +\infty }{lim\; inf}{\int }_{|x|\le a(t_n)}|u(x,t_n){|}^{2}dx\ge \underset{n\to +\infty }{lim\; inf}{\int }_{|x|<r{\rho }_n}{|u(x,t_n)|}^{2}dx\hfill \\ \hfill & \ge \underset{n\to +\infty }{lim\; inf}{\int }_{|x|\le R}{\rho }_n^N|u({\rho }_{n}x,t_n){|}^{2}dx\ge \underset{n\to +\infty }{lim\; inf}{\int }_{|x|\le R}{|v(t_{n}x)|}^{2}dx\hfill \\ \hfill & \ge \underset{n\to +\infty }{lim\; inf}{\int }_{|x|\le R}{|V(x)|}^{2}dx\hfill \end{array}$$

(41)

for every $r>0$, which means that

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; inf}{\int }_{|x|\le a(t_n)}|u(x,t_n){|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{|V(x)|}^{2}dx.\end{array}$$

Because the sequence ${\{t_n\}}_{j=1}^{+\infty }$ is arbitrary, it follows from (41) that

$$\begin{array}{c}\hfill \underset{t\to T}{lim\; inf}{\int }_{|x|<a(t)}{|u(x,t)|}^{2}dx\ge {\int }_{{\mathbb{R}}^N}{|Q(x)|}^{2}dx.\end{array}$$

This completes the proof. □

Remark 7.

Comparing Theorem 2 and Theorem 5, we can find the effect of the inhomogeneous on the $L^2$ concentration phenomenon. Affected by the singular coefficients ${|x|}^{-b}$, when inhomogeneous term dominate the equation, the mass of the finite time blow-up solution can only be concentrated near the origin.

Remark 8.

Unfortunately, we only know the existence but not the uniqueness of the solution to the following equation

$$-{(-\Delta )}^{s}Q-Q+{|x|}^{-b}{|Q|}^{\frac{4s-2b+2N}{N}}Q=0.$$

If we can obtain the uniqueness of radial solution of above problem, we can study the limiting profile of radial blow-up solutions of problem (25) similar to Theorem 3.

Remark 9.

When $p=\frac{4s-2b}{N}$ and $q<\frac{4s}{N}$, it is possible that q is larger than the critical exponent $\frac{4s-2b}{N}$ of the inhomogeneous term, but this does not affect the $L^2$ concentration phenomenon of the solution at the blow-up time.