Existence and Non-existence of Global Solutions for a Nonlocal Choquard–Kirchhoff Diffusion Equations in $\mathbb {R}^{N}$

Abstract

In this paper, we investigate the local existence, global existence, and blow-up of solutions to the Cauchy problem for Choquard–Kirchhoff-type equations involving the fractional p-Laplacian. As a particular case, we study the following initial value problem

$$\begin{aligned}\left\{ \begin{array}{llc} u_{t}+M\left( \Vert u\Vert ^{p}\right) [(-\Delta )^{s}_{p}u+V(x)|u|^{p-2}u]=\left( \int _{\mathbb {R}^{N}}\frac{|u|^{q}}{|x-y|^{\mu }}\,dy\right) |u|^{q-2} u&{} \text {in}\ &{}\mathbb {R}^{N}\times (0, +\infty ), \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\mathbb {R}^{N} , \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} \Vert u\Vert =\left( [u]^{p}_{s,p}+\int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx\right) ^{1/p}, \end{aligned}$$

$s\in (0,1)$, $N>ps$, $p,q> 2$, $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian, $u_{0} :\mathbb {R}^{N}\rightarrow [0, +\infty )$ is the initial function, $M :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ is a continuous function given by $M(\sigma )=\sigma ^{\theta -1}$, $\theta \in [1, N/(N-sp))$ and $V :\mathbb {R}^{N}\rightarrow \mathbb {R}^{+}$ is the potential function. Under some appropriate conditions, the well-posedness of nonnegative solutions for the above Cauchy problem is established by employing the Galerkin method. Moreover, the asymptotic behavior of global solutions is investigated under some assumptions on the initial data. We also establish upper and lower bounds for the blow-up time.

1 Introduction and the Main Results

The aim of this paper is to discuss the global well-posedness, asymptotic behavior and blow-up phenomena to the following fractional Choquard–Kirchhoff-type parabolic equations

$$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+M\left( \Vert u\Vert ^{p}\right) [(-\Delta )^{s}_{p}u+V(x)|u|^{p-2}u]=(\mathcal {K}_{\mu }*|u|^{q})|u|^{q-2} u&{} \text {in}\ &{} \mathbb {R}^{N}\times (0, +\infty ) , \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\mathbb {R}^{N} , \end{array}\right. \end{aligned}$$

(1.1)

where $p,q> 2$, $M :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ is a continuous function given by $M(\sigma )=\sigma ^{\theta -1}$, $\theta \in [1, N/(N-sp))$, $s\in (0,1)$, $N> sp$, $\mu \in (0, N)$, and $V:\mathbb {R}^{N}\rightarrow \mathbb {R}^{+}$ is a scalar potential. Hereafter $\mathcal {K}_{\mu }(x)=|x|^{-\mu }$,

$$\begin{aligned} \Vert u\Vert =\left( [u]^{p}_{s,p}+\int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx\right) ^{1/p}, \;\;\; [u]_{s,p}=\left( \int _{\mathbb {R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy\right) ^{1/p}. \end{aligned}$$

$(-\Delta _{p})^{s}$ is the fractional p-Laplacian which, up to a normalization constant, is defined for each $x\in \mathbb {R}^{N}$ as

$$\begin{aligned} (-\Delta )_{p}^{s}\varphi (x)=2 \lim _{\epsilon \rightarrow 0^{+}}\int _{\mathbb {R}^{N}\backslash B_{\epsilon }(x)}\frac{|\varphi (x)-\varphi (y)|^{p-2}(\varphi (x)-\varphi (y))}{|x-y|^{N+sp}}\,dy \end{aligned}$$

(1.2)

for all $\varphi \in C^{\infty }_{0}(\mathbb {R}^{N})$. Here $B_{\epsilon }(x)$ denotes the ball in $\mathbb {R}^{N}$ centered at x with radius $\epsilon$.

In the last few decades, great attention has been paid to study problems involving fractional Laplacian. The interest in studying such problems was stimulated by their applications in continuum mechanics, minimal surfaces, conversation laws, population dynamics, image processing, finance, and many others, see for example [1,2,3] and the references therein. In particle the study of parabolic equations involving fractional p-Laplacian like (1.1) are important in many field of sciences, such as biology, geophysics, as well as in Riemannian geometry in the so-called scalar curvature problem on the sphere $\mathbb {S}^{N}$, for more details see [4,5,6]. From the mathematical point of view, the difficulty in solving problem like (1.1) is due to the lack of compactness which caused by the invariant action of the conformal group, or one of its subgroups, we refer to [7] for details. When $\mu =p=q=2$ and $V \equiv M\equiv 1$, the stationary equation related to (1.1) becomes the well-known Choquard or nonlinear Schrödinger-Newton equation

In the case when $N=3$, equation $(E_{1})$ was first introduced in 1954 by Pekar [8]. In 1996, Penrose [9] used equation $(E_{1})$ in a different context as a model in self-gravitating matter. The literature on equations of the type $(E_{1})$ is very large and rich, so here we just list some papers where the authors studied the existence and multiplicity of solutions for $(E_{1})$, see for example [10,11,12,13] and the references therein. If $s\rightarrow 1^{-}$, $p=2$ and $V\equiv 0$, the homogeneous equation related to (1.1) reduces to the following equation

As far as we know, the first and the only result on the global existence and nonexistence of solutions of $(E'_{1})$ was obtained by Gobbino [14] in a more abstract setting. In this work the author classified the existence and nonexistence of solutions of $(E'_{1})$ into the following cases :

• If $u_{0}\in D(A^{1/2})$ and $M\left( \Vert \nabla u_{0}\Vert _{2}^{2}\right) \ne 0$, then there exists at least one global solution.

• If $u_{0} \in D(A^{\beta })$ with $\beta$ small enough, then there is no solution.

Equation $(E'_{1})$ is related to the parabolic analogue of the Kirchhoff equation

This equation was first introduced by Gustav Robert Kirchhoff in 1876, which describes the movement of an elastic string that is constrained at the extrema, taking into account a possible growth of the tension of the vibrating string in view of its extension.

Recently, many authors investigated the existence of local and global solutions for this abstract initial value problem

$$\begin{aligned} u''(t)+M\left( \Vert A^{\frac{1}{2}}u(t)\Vert ^{2}\right) Au(t)=0, \;\;u(0)=u_{0}, \;u'(0)=u_{1}, \end{aligned}$$

(P)

where A is a nonnegative self-adjoint linear operator on a Hilbert space H with dense domain D(A). Powers of the operator A are just defined by the spectral operator, that is

$$\begin{aligned} A^{s}u :=\sum _{k=0}^{\infty }\lambda ^{2s}_{k}u_{k}e_{k}, \;\;\forall s \ge 0, \;\;\forall u\in D(A^{s}). \end{aligned}$$

(1.3)

The local and global existence of solutions of (P) has been proved under different assumptions on the initial data $(u_{0}, u_{1})$ and on the nonlinear function M, see for example the works by Ghisi and Gobbino [15,16,17] and the references cited therein. It is important to point out here when $p=2$ and $s\in (0,1)$ the fractional Laplace operator $(-\Delta )^{s}$ defined in (1.2) and the spectral operator $A^{s}$ given in (1.3) are completely different, we refer the reader to the monograph [18] for a comparison between these operators.

In the whole space $\mathbb {R}^{N}$, Papadopoulos and Stavarakakis [19] investigated the global existence and blow-up of solutions for the following nonlocal quasilinear hyperbolic problem of Kirchhoff type

where $N\ge 3$, $\delta \ge 0$ and $\rho (x)=\left( \phi (x)\right) ^{-1}$ is a positive function lying in $L^{N/2}(\mathbb {R}^{N})\cap L^{\infty }(\mathbb {R}^{N}).$ [20, 21]

It is worth motioning that the global existence and blow-up of solutions for the parabolic equations of Kirchhoff type involving the classical Laplacian operator have been widely studied by many authors. For instance, we refer to [22,23,24,25] and the references therein for the setting of bounded domains. Concerning the global existence, asymptotic behavior and blow-up of solutions for hyperbolic equations of Kirchhoff type involving the fractional Laplacian in the case of bounded domain, we refer the reader to some recent results obtained in [26, 27]. However, to the author’s best knowledge there is no result on the global existence and blow-up of solutions for parabolic and hyperbolic equations of Kirchhoff type involving fractional Laplacian operator in whole $\mathbb {R}^{N}$. Recently, Fiscella and Valdinoci [28] proposed a fractional counterpart to the Kirchhoff operator which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. We recall that Kirchhoff problems, with Kirchhoff function M, are said to be non-degenerate if $M(0)> 0$, and degenerate if $M(0)=0$. Xiang, R$\breve{\text {a}}$dulescu and Zhang [5] considered the following diffusion model of Kirchhoff-type

where $\Omega$ is a bounded domain in $\mathbb {R}^{N}$ and $(-\Delta )^{s}$ is the fractional Laplacian with $0< s< 1< p<\infty$. Under some appropriate conditions the authors obtained a nonnegative local weak solution of $(E_{2})$ by using the Galerkin method. Moreover, they proved also an estimate for the lower and upper bounds of the blow-up time. Leter, by combining the Galerkin method with potential well theory, Ding and Zhou [29] investigated the global existence and blow-up of solutions for problem $(E_{2})$. Pucci et al. [30] studied the following anomalous diffusion model of Kirchhoff type

where $\Omega \subset \mathbb {R}^{N}$ is a bounded domain, $f\in L_{loc}^{2}(\mathbb {R}^{+}_{0}, L^{2}(\Omega ))$ and $(-\Delta )_{p}^{s}$ is the fractional p-Laplacian. By using the sub-differential approach, the authors established the well-posedness of solutions for problem $(E_{3})$. Moreover, the large time behavior and extinction of solutions are also investigated. With the help of potential well theory, Fu and Pucci [31], studied the existence of global weak solutions and established the vacuum isolating and blow-up of strong solutions for the following class of problem

where $s\in (0,1)$, $N>2s$ and $2< p\le 2_{s}^{*}=2N/(N-2s).$

To the best of our knowledge, there is no result on global existence and asymptotic behavior as well as blow-up of solutions in finite time for initial problem (1.1). There is no doubt we encounter serious difficulties because of the lack of compactness, Hardy–Littlewood–Sobolev nonlinearity as well as the degenerate nature of the Kirchhoff coefficient.

Inspired by the above works, especially by [5, 14, 29], we study the well-posedness and asymptotic behavior as well as blow-up of solutions in finite time for initial problem (1.1). Moreover, we give an estimate for the lower and upper bounds of the blow-up time. We stress that these results are new even in the case of classical Laplacian where $M\equiv 1$.

In order to present the main results of this paper, let us recall some results related to the fractional Sobolev space $W^{s,p}(\mathbb {R}^{N})$ (see [18, 32]). Let $0< s<1<p<\infty$ be real numbers. The fractional Sobolev space $W^{s,p}(\mathbb {R}^{N})$ is defined as follows :

$$\begin{aligned} W^{s,p}(\mathbb {R}^{N})=\left\{ u \in L^{p} (\mathbb {R}^{N})\,:\int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy< \infty \right\} , \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{W^{s,p}(\mathbb {R}^{N})}=\left( \Vert u\Vert ^{p}_{p}+\int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy\right) ^{1/p}. \end{aligned}$$

As it is well known $(W^{s,p}(\mathbb {R}^{N}), \Vert \,.\,\Vert _{W^{s,p}(\mathbb {R}^{N})})$ is a reflexive separable Banach space and the embedding $W^{s, p}(\mathbb {R}^{N})\hookrightarrow L^{q}(\mathbb {R}^{N})$ is continuous for any $q\in [p, p_{s}^{*}]$, namely there exists a positive constant $S_{q}$ such that

$$\begin{aligned} \Vert u\Vert _{q}\le S_{q}\Vert u\Vert _{W^{s,p}(\mathbb {R}^{N})}. \end{aligned}$$

Here $p^{*}_{s}$ is the critical exponent defined as $p^{*}_{s}=\frac{Np}{N-sp}$, see [32] for more details. In order to obtain the existence of weak solutions for (1.1), we consider the subspace of $W^{s, p}(\mathbb {R}^{N})$

$$\begin{aligned} W=\left\{ u\in W^{s, p}(\mathbb {R}^{N})\,:\int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx< \infty \right\} , \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert =\left( \Vert V^{1/p}u\Vert ^{p}_{p}+\int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,dxdy\right) ^{1/p}. \end{aligned}$$

Throughout the paper we assume the following hypotheses on V :

A typical example of V is given by $V(x)=\exp (\rho |x|^{2})$ for all $\rho >0$.

The energy functional associated with initial value problem (1.1) is given by

$$\begin{aligned} E(u)=\frac{1}{\theta p}\Vert u\Vert ^{\theta p}-\frac{1}{2q}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx,\;\;\;\;u\in W. \end{aligned}$$

(1.4)

We define the Nehari functional by

$$\begin{aligned} I(u)=\Vert u\Vert ^{\theta p}-\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx,\;\;\;\;u\in W. \end{aligned}$$

(1.5)

In view of (1.4) and (1.5), it is easy to see that

$$\begin{aligned} E(u)=\left( \frac{1}{\theta p}-\frac{1}{2q}\right) \Vert u\Vert ^{\theta p}+\frac{1}{2q}I(u). \end{aligned}$$

(1.6)

Let

$$\begin{aligned} d=\inf _{u\in \mathcal {N}}E(u), \end{aligned}$$

(1.7)

where $\mathcal {N}$ is the Nehari manifold, which is defined by

$$\begin{aligned} \mathcal {N}=\{u\in W\backslash \{0\}\,: I(u)=0\}. \end{aligned}$$

(1.8)

We shall show in Lemma 3.3 below that

$$\begin{aligned} d\ge \left( \frac{2q-\theta p}{2\theta qp}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}, \end{aligned}$$

(1.9)

where $\Lambda _{\nu }$ denotes the best constant in the embedding $W\hookrightarrow L^{\nu } (\mathbb {R}^{N})$ for all $\nu \in [2,p^{*}_{s}]$, i.e,

$$\begin{aligned} \Lambda _{\nu } =\inf \left\{ \frac{\Vert u\Vert }{\Vert u\Vert _{\nu }}\,:u\in W\backslash \{0\}\right\} . \end{aligned}$$

(1.10)

Clearly, $\Lambda _{\nu }>0.$ Before stating the main results of this paper, let us give the definition of the weak solutions for initial value problem (1.1).

Definition 1.1

We say that $u\in L^{\infty }(0,T; W)$ is a positive weak solution of problem (1.1) if, $u_{t} \in L^{2}(0, T; L^{2}(\mathbb {R}^{N}))$ and the following equalities hold

• (1) $\int _{\mathbb {R}^{N}}u_{t}v\,dx+\Vert u\Vert ^{p(\theta -1)} \langle u,v\rangle =\int _{\mathbb {R}^{N}} (\mathcal {K}_{\mu }*u^{q})u^{q-1} uv\,dx$ for each $v\in W$ and a.e time $0\le t \le T$, and

• (2) $u(0)=u_{0}$.

where

$$\begin{aligned} \langle u,v\rangle= & {} \int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\,dxdy\\&+\int _{\mathbb {R}^{N}} V|u|^{p-2}uv\,dx. \end{aligned}$$

Our first result reads as follows :

Theorem 1.1

Suppose that $0\le u_{0}\in W$ and the following condition holds :

$$\begin{aligned}&2<qr<2q< \min \left\{ \theta p+\frac{4p^{*}_{s}-2r\theta p}{rp^{*}_{s}}, \frac{(2-r)p^{*}_{s}}{2}\right\} , \nonumber \\&r=\frac{2N}{2N-\mu }, \;\;\;p^{*}_{s}=\frac{Np}{N-sp}. \end{aligned}$$

(1.11)

Then there exists $T>0$ such that problem (1.1) admits at least a nontrivial, nonnegative weak solution for all $t\in (0, T]$. Moreover, for a.e. $t\in [0,T]$,

$$\begin{aligned} \int _{0}^{t}\Vert u_{s}(s)\Vert ^{2}_{2}\,ds+E(u(t))\le E(u_{0}). \end{aligned}$$

(1.12)

Remark 1

From the regularity of weak solutions stated in Definition 1.1 and [33, Proposition 1.2] we infer that $u\in C([0,T], L^{2}(\mathbb {R}^{N}))$. Therefore, the initial condition (2) in Definition 1.1 exists and makes sense.

Before introducing the second result, let us give the following definition :

Definition 1.2

(Maximal existence time) Let u(t) be a solution of problem (1.1). We define the maximal existence time $T_{\max }$ of u as follows :

$$\begin{aligned} T_{\max }=\sup \{t>0\, : u=u(t)\;\text { exists on }\, [0, T]\}. \end{aligned}$$

• (1) If $T_{\max }<\infty$ we say that the solution of (1.1) blows up and $T_{\max }$ is the blow up time.

• (2) If $T_{\max }=\infty$, we say that the solution is global.

The proof of the following Theorem 1.2 relies on the potential well method which was introduced by Sattinger in [34]; see [25, 35, 36] and the references therein for some results on global existence of solutions.

Theorem 1.2

Assume that $0\le u_{0}\in W$ and the following conditions hold true :

$$\begin{aligned} \theta p< 2q< p^{*}_{s}, \end{aligned}$$

(1.13)

and

$$\begin{aligned} E(u_{0})< d \quad \text { and } \quad I(u_{0})> 0. \end{aligned}$$

(1.14)

Then the positive weak solution $u=u(t)$ of initial value problem (1.1) exists globally. Moreover,

$$\begin{aligned} \int _{0}^{t}\Vert u_{s}(s)\Vert ^{2}_{2}\,ds+E(u(t))\le E(u_{0}), \;\;\text { a.e. }\,t\in [0, +\infty ). \end{aligned}$$

(1.15)

Remark 2

Let us discuss the initial value for which the condition (1.14) is satisfied.

• (1) If $0< E(u_{0})<d$ and $I(u_{0})\ge 0$ then we have $I(u_{0})>0$. Indeed, assume that $I(u_{0})=0$. From (1.6) we observe that $u_{0}\ne 0$. Thus, $u_{0} \in \mathcal {N}$, then by the definition of d in (1.7), we infer that $E(u_{0})\ge d$, contradiction.

• (2) If $E(u_{0})=0$ and $I(u_{0})\ge 0$ then we have $u_{0}=0$. In fact, if we assume $u_{0}\ne 0$. Then from (1.6) it follows that $I(u_{0})=0$. This gives $u_{0}\in \mathcal {N}$ and $0=E(u_{0}) \ge d>0$, contradiction by (1.9).

• (3) By (1.6), clearly the case $E(u_{0})<0$ and $I(u_{0})\ge 0$ can not occur.

Furthermore, we have the following corollary to Theorem 1.2.

Corollary 1.3

Let $0\le u_{0}\in W$ and (1.13) holds. Assume that

$$\begin{aligned} E(u_{0})\le d \quad \text { and } \quad I(u_{0})\ge 0. \end{aligned}$$

(1.16)

Then initial value problem (1.1) admits a global weak solution.

The following theorem shows the asymptotic behavior of global solutions of initial value problem (1.1).

Theorem 1.4

Suppose the assumptions made in Theorem 1.2 are satisfied. Assume that $\epsilon \in (0,s)$ and

$$\begin{aligned} E(u_{0})< \left( \frac{2q-\theta p}{2q\theta p}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\frac{\theta p}{2q-\theta p}}. \end{aligned}$$

(1.17)

Then, there holds

$$\begin{aligned}&\Vert u(t)\Vert _{W^{s-\epsilon , p}(\mathbb {R}^{N})} \le \omega \left( \frac{\theta p}{1+\chi (\theta p-1)t}\right) ^{\frac{\beta (1-\eta )}{\theta p-1}}, \, \;\; \forall \eta ,\beta \in (0,1),\;\;\forall t \ge 0.\nonumber \\ \end{aligned}$$

(1.18)

where $\omega = \Lambda _{p^{*}_{s}}^{(\beta -1)(1-\eta )}\left( \frac{2q\theta p}{2q-\theta p}\right) ^{\frac{\eta +(1-\eta ) (1-\beta )}{\theta p}}$ and $\chi =\left( 1-C(N,\mu , r)\Lambda _{rq}^{-2q}\left( \frac{2q\theta p}{2q-\theta p}E(u_{0})\right) ^{\frac{2q-\theta p}{\theta p}}\right)$.

The following theorem shows that the local solution obtained in Theorem 1.1 can not be extended globally in time.

Theorem 1.5

Suppose that $0\le u_{0} \in W$, $\theta \in (2, N/(N-sp))$ and the following conditions hold :

$$\begin{aligned}&\theta p<qr<2q< \min \left\{ \theta p+\frac{4p^{*}_{s}-2r\theta p}{rp^{*}_{s}}, \frac{(2-r)p^{*}_{s}}{2}\right\} , \nonumber \\&r=\frac{2N}{2N-\mu }, \;\;\;p^{*}_{s}=\frac{Np}{N-sp}, \end{aligned}$$

(1.19)

$$\begin{aligned}&E(u_{0})< 0. \end{aligned}$$

(1.20)

Then the local weak solution u(t) obtained in Theorem 1.1 belows up in finite time T, where T satisfies that

$$\begin{aligned} \frac{2^{\frac{2\theta ^{2} p^{2}}{\theta p-2q(1-\gamma )}}\Lambda ^{\frac{2q(1-\gamma )\theta p}{\theta p-2q(1-\gamma )}}_{p^{*}_{s}}}{(\alpha -1)\Vert u_{0}\Vert _{2}^{2(\alpha -1)}}\le T\le \frac{8\Vert u_{0}\Vert ^{2}_{2}(\theta p-1)}{-\theta pE(u_{0}) (\theta p-4)^{2}}. \end{aligned}$$

with $\alpha =\frac{\theta pq\gamma }{\theta p-2q(1-\gamma )}> 1$.

Remark 3

Notice that, due to the fact that $2r\theta p< 4\theta p<4p^{*}_{s}$ the condition (1.19) exists. Hence, from (1.19) we observe that the condition (1.11) still holds. Therefore, in view of Theorem 1.1 there exists a local weak solution for problem (1.1) satisfies the energy inequality (1.12).

Remark 4

The results presented in this paper can be easily extended to more general Choquard–Kirchhoff-type equations. For example, with the same technique, we can deal with nonlinearities $M :[0, +\infty )\rightarrow [0, +\infty )$ of class $C^{1}$ such that

$$\begin{aligned} c_{1}r^{\theta -1} \le M(r) \le c_{2}r^{\theta -1} \end{aligned}$$

for suitable positive constants $c_{1}$ and $c_{2}$. However, this generality only complicates proofs without bringing any new idea.

The remaining part of the paper is organized as follows. Section 2, contains some preliminary results, which are required in the proof of the main results. In section 3, we shall present some properties involving the functional E restricted to the Nehari manifold $\mathcal {N}$. In section 4, we establish local existence of solutions by using the Galerkin method. In section 5, under some conditions on the initial data, we show the global existence of solutions for (1.1). Furthermore, we give a decay estimate for these global solutions in fractional Sobolev spaces for large time. Finally, in section 6, finite time blow-up of weak solutions of (1.1) is proved, in addition to this we give an estimate for the upper and lower bounds of the blow-up time.

Throughout the paper $c,c_{i}, C, C_{i},$ $i=1,2,\ldots$ denote positive constants which may vary from line to line, but are independent of terms that take part in any limit process and we use the notation $\Vert \,.\,\Vert _{p}$ for the standard $L^{p}(\mathbb {R}^{N})-$norm. Furthermore we use “$\rightarrow$”, “$\rightharpoonup$” and “$\rightharpoonup ^{*}$” to denote the strong convergence, weak convergence and weak star convergence respectively.

2 Functional Framework

In this section, we give some technical results that will be used in the next sections.

Lemma 2.1

Let $(V_{1})$ and $(V_{2})$ hold. Then the embedding $W \hookrightarrow L^{q}(\mathbb {R}^{N})$ is continuous for all $q\in [2, p^{*}_{s}].$

Proof

We have

$$\begin{aligned} \int _{\mathbb {R}^{N}}|u|^{2}\,dx=\int _{\mathbb {R}^{N}}\frac{1}{V^{\frac{2}{p}}(x)}V^{\frac{2}{p}}(x)|u|^{2}\,dx. \end{aligned}$$

Since $p> 2$ from condition $(V_{2})$ and the Hölder inequality we deduce

$$\begin{aligned} \int _{\mathbb {R}^{N}}|u|^{2}\,dx\le \left( \int _{\mathbb {R}^{N}}\frac{1}{V^{\frac{2}{p-2}}(x)}\,dx\right) ^{(p-2)/2}\left( \int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx\right) ^{2/p}\le C \Vert u\Vert ^{2}. \end{aligned}$$

On the other hand, by $(V_{1})$ we have $W\hookrightarrow W^{s,p}(\mathbb {R}^{N})\hookrightarrow L^{p^{*}_{s}}(\mathbb {R}^{N})$. Then $2< q<p^{*}_{s}$ and $\theta \in (0,1)$ satisfy the following equation

$$\begin{aligned} \frac{1}{q}=\frac{\theta }{p^{*}_{s}}+\frac{1-\theta }{2}. \end{aligned}$$

By the Hölder inequality, we have

$$\begin{aligned} \Vert u\Vert _{q}\le \Vert u\Vert ^{1-\theta }_{2}\Vert u\Vert ^{\theta }_{p^{*}_{s}}\le C \Vert u\Vert . \end{aligned}$$

Hence this completes the proof. $\square$

From now on, $B_{R}(0)$ denotes the ball in $\mathbb {R}^{N}$ of center zero and radius $R>0$.

Lemma 2.2

Assume that $(V_{1})$ and $(V_{2})$ hold. Then the embedding $W\hookrightarrow L^{q}(\mathbb {R}^{N})$ is compact for all $q\in [2, p^{*}_{s}).$

Proof

It is well know that the embedding $W^{s,p}(\mathbb {R}^{N})\hookrightarrow L^{q}(B_{R}(0))$ is compact for all $q\in [2, p^{*}_{s}).$ Therefore $W\hookrightarrow L^{q}(B_{R}(0))$ is compact for all $q\in [2, p^{*}_{s}).$ Since $\frac{1}{V} \in L^{\frac{2}{p-2}}(\mathbb {R}^{N})$, for any $\varepsilon >0$ there exists $R_{1}>0$ such that

$$\begin{aligned} \left( \int _{\mathbb {R}^{N}\backslash B_{R}(0)}\frac{1}{V^{\frac{2}{p-2}}(x)}\,dx\right) ^{(p-2)/p}<\varepsilon , \;\;\text {for all }\,R\ge R_{1}. \end{aligned}$$

(2.1)

Let $\{u_{m}\}_{m}$ be a bounded sequence in W. Then, for any $\varepsilon >0$ there exists $m_{0}>0$ such that

$$\begin{aligned} \int _{B_{R_{1}}(0)}|u_{m}-u|^{2}\,dx< \varepsilon , \;\;\text {for all}\; m\ge m_{0}. \end{aligned}$$

(2.2)

Combining (2.1) and (2.2), for all $m\ge m_{0}$ we have

$$\begin{aligned} \int _{\mathbb {R}^{N}}|u_{m}-u|^{2}\,dx= & {} \int _{B_{R_{1}}(0)}|u_{m}-u|^{2}\,dx+\int _{\mathbb {R}^{N}\backslash B_{R_{1}}(0)}\frac{1}{V^{\frac{2}{p}}(x)}V^{\frac{2}{p}}(x)|u_{m}-u|^{2}\,dx\\\le & {} \varepsilon + C_{1} \left( \int _{\mathbb {R}^{N}\backslash B_{R_{1}}(0)}\frac{1}{V^{\frac{2}{p-2}}(x)}\,dx\right) ^{p-2/p}\le (1+C_{1})\varepsilon . \end{aligned}$$

This combined with $2< q< p^{*}_{s}$ yields

$$\begin{aligned} \Vert u_{m}-u\Vert _{q}\le \Vert u_{m}-u\Vert ^{\theta }_{p^{*}_{s}}\Vert u_{m}-u\Vert ^{1-\theta }_{2}\le C_{2}\varepsilon ^{1-\theta }, \;\forall \theta \in (0,1),\;\forall \, m\ge m_{0}. \end{aligned}$$

Thus $\Vert u_{m}-u\Vert _{q}\rightarrow 0$, as $m\rightarrow \infty$. Hence the proof is now complete. $\square$

Throughout this paper, we shall assume that q satisfies

$$\begin{aligned} 2< q< \frac{p(N-\frac{\mu }{2})}{N-sp}. \end{aligned}$$

(1.3)

In what follows we recall the so-called Hardy–Littlewood–Sobolev inequality, see [37, Theorem 4.3]. Hereafter, $W'$ denotes the dual space of W.

Theorem 2.3

Assume that $1< r, j< \infty$, $0< \mu < N$ and

$$\begin{aligned} \frac{1}{r}+\frac{1}{j}+\frac{\mu }{N}=2. \end{aligned}$$

Then there exists $C(N,\mu , j,r)> 0$ such that

$$\begin{aligned} \int _{\mathbb {R}^{2N}} \frac{|u(y)|.|u(x)|}{|x-y|^{\mu }}\,dxdy\le C(N,\mu , j,r) \Vert u\Vert _{j}\Vert u\Vert _{r}. \end{aligned}$$

From this Theorem, one can observe that the operator $H: W\rightarrow W'$ defined by

$$\begin{aligned} \langle H(u), v\rangle =\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q-2}uv\,dx. \end{aligned}$$

for all $u,v\in W$ is well defined. Indeed, clearly, for all $u\in W$, H(u) is a linear map. Using the Hardy–Littlewood–Sobolev inequality for $j=r=\frac{2N}{2N-\mu }$ we obtain

$$\begin{aligned} \int _{\mathbb {R}^{2N}} \frac{|u(y)|^{q}.|u(x)|^{q-1}.|v(x)|}{|x-y|^{\mu }}\,dxdy\le C(N,\mu , r) \Vert |u|^{q}\Vert _{r}\Vert |u|^{q-1}|v|\Vert _{r}. \end{aligned}$$

Note that by (1.3) we have $2< qr<p^{*}_{s}$. In view of the Hölder inequality and the continuous embedding $W\hookrightarrow L^{\kappa }(\mathbb {R}^{N})$ for all $\kappa \in [2,p^{*}_{s}]$ we deuce

$$\begin{aligned} \int _{\mathbb {R}^{2N}} \frac{|u(y)|^{q}.|u(x)|^{q-1}.|v(x)|}{|x-y|^{\mu }}\,dxdy \le C(N,\mu , r) \Vert u\Vert ^{2q-1}\Vert v\Vert . \end{aligned}$$

Therefore, for $v\in W$ with $\Vert v\Vert \le 1$ we obtain

$$\begin{aligned} \Vert H(u)\Vert _{W'}\le C(N,\mu , r) \Vert u\Vert ^{2q-1} . \end{aligned}$$

Lemma 2.4

[38, Proposition 4.7.12] Let $\kappa \in (1, \infty )$. Assume $\{w_{m}\}$ is a bounded sequence in $L^{\kappa }(\mathbb {R}^{N})$ and converges almost everywhere to w. Then $w_{m}\rightharpoonup w$ in $L^{\kappa }(\mathbb {R}^{N})$.

Lemma 2.5

(Brezis-Lieb lemma) Let $\kappa \in (1, \infty )$. Assume $\{w_{m}\}$ is a bounded sequence in $L^{\kappa }(\mathbb {R}^{N})$ and converges almost everywhere to w. Then for any $t\in [1,\kappa ]$

• (1) $\lim \limits _{m\rightarrow \infty } \int _{\mathbb {R}^{N}}||w_{m}|^{t-2}w_{m}h-|w_{m}-w|^{t-2}(w_{m}-w)h-|w|^{t-2}wh|^{\frac{\kappa }{t}}\,dx=0$, for all $h\in L^{\kappa }(\mathbb {R}^{N}),$

• (2) $\lim \limits _{m\rightarrow \infty } \int _{\mathbb {R}^{N}}||w_{m}|^{t}-|w_{m}-w|^{t}-|w|^{t}|^{\frac{\kappa }{t}}\,dx=0.$

Proof

Using the Young inequality, for any $\varepsilon > 0$ there exist $C(\varepsilon ), C_{1}>0$ such that for all $a,b\in \mathbb {R}$ we have

$$\begin{aligned} \big ||a+b|^{t-2}(a+b)h-|a|^{t-2}ah\big |^{\frac{\kappa }{t}}\le C_{1}|h|^{\kappa }+C(\varepsilon )|b|^{\kappa }+\varepsilon |a|^{\kappa }. \end{aligned}$$

Taking $a=w_{m}-w$ and $b=w$ and using the above inequality, yields

$$\begin{aligned} f_{m,\varepsilon }&=(||w_{m}|^{t-2}w_{m}h-|w_{m}-w|^{t-2}(w_{m}-w)h-|w|^{t-2}wh|^{\frac{\kappa }{t}}-\varepsilon |w_{m}-w|^{\kappa })^{+}\\&\le (1+C(\varepsilon ))|w|^{\kappa }+C_{2}|h|^{\kappa }. \end{aligned}$$

Note that, according to the above assumptions we have $(1+C(\varepsilon ))|w|^{\kappa }+C_{2}|h|^{\kappa }\in L^{1}(\mathbb {R}^{N})$ and $f_{m,\varepsilon }\rightarrow 0$ a.e in $\mathbb {R}^{N}$. Thus the Lebesgue dominated convergence theorem implies

$$\begin{aligned} \int _{\mathbb {R}^{N}}f_{m,\varepsilon }\,dx\rightarrow 0\;\text {as}\; m\rightarrow \infty . \end{aligned}$$

Therefore, we get

$$\begin{aligned} ||w_{m}|^{t-2}w_{m}h-|w_{m}-w|^{t-2}(w_{m}-w)h-|w|^{t-2}wh|^{\frac{\kappa }{t}}\le f_{m,\varepsilon }+\varepsilon |w_{m}-w|^{\kappa }, \end{aligned}$$

which implies

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{\mathbb {R}^{N}}||w_{m}|^{t-2}w_{m}h-|w_{m}-w|^{t-2}(w_{m}-w)h-|w|^{t-2}wh|^{\frac{\kappa }{t}}\,dx\le c\varepsilon . \end{aligned}$$

where $c=sup_{m}\Vert w_{m}-w\Vert ^{\kappa }_{\kappa }< \infty$. Further letting $\varepsilon \rightarrow 0$, we obtain the desired result. In a similar manner, we conclude that (2) holds. $\square$

Throughout the paper without further mentioning, we put $r=\frac{2N}{2N-\mu }$.

Lemma 2.6

Let $\{u_{m}\}_{m}$ be a bounded sequence in $L^{rq}(\mathbb {R}^{N})$ such that $u_{m}\rightarrow u$ a.e in $\mathbb {R}^{N}$ as $m\rightarrow \infty$. Then for any $w\in L^{rq}(\mathbb {R}^{N})$ we have

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{m}|^{q})|u_{m}|^{q-2}u_{m}w\,dx \rightarrow \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q-2}uw\,dx. \end{aligned}$$

Proof

Denote $v_{m}=u_{m}-u$ and observe that

$$\begin{aligned}&\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{m}|^{q})|u_{m}|^{q-2}u_{m}w\,dx \nonumber \\&\quad = \int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q}-|v_{m}|^{q})](|u_{m}|^{q-2}u_{m}w-|v_{m}|^{q-2}v_{m}w)\,dx\nonumber \\&\quad \quad + \int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q}-|v_{m}|^{q})]|v_{m}|^{q-2}v_{m}w\,dx \nonumber \\&\quad \quad +\int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q-2}u_{m} w-|v_{m}|^{q-2}v_{m} w)]|v_{m}|^{q}\,dx \nonumber \\&\quad \quad +\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|v_{m}|^{q})|v_{m}|^{q-2}v_{m}w\,dx \end{aligned}$$

(1.4)

Applying Lemma 2.5 with $t=q$ and $\kappa =rq$, we find

$$\begin{aligned} |u_{m}|^{q}-|v_{m}|^{q}\rightarrow |u|^{q}\;\;\text {in}\;L^{r}(\mathbb {R}^{N}), \end{aligned}$$

and

$$\begin{aligned} |u_{m}|^{q-2}u_{m}w-|v_{m}|^{q-2}v_{m}w\rightarrow |u|^{q-2}uw\;\;\text {in}\;L^{r}(\mathbb {R}^{N}). \end{aligned}$$

The Hardy–Littlewood–Sobolev inequality ensures that

$$\begin{aligned} \left\{ \begin{array}{cc} \mathcal {K}_{\mu }*(|u_{m}|^{q}-|v_{m}|^{q})\rightarrow \mathcal {K}_{\mu }*|u|^{q} &{} \\ &{} \text {in}\;\;L^{\frac{2N}{\mu }}(\mathbb {R}^{N}).\\ \mathcal {K}_{\mu }*(|u_{m}|^{q-2}u_{m}w-|v_{m}|^{q-2}v_{m}w)\rightarrow \mathcal {K}_{\mu }*(|u|^{q-2}uw) \end{array}\right. \end{aligned}$$

(1.5)

In view of Lemma 2.4, we have

$$\begin{aligned} |u_{m}|^{q-2}u_{m}w\rightharpoonup |u|^{q-2}uw, \;\;\; |v_{m}|^{q}\rightharpoonup 0, \;\;|v_{m}|^{q-2}v_{m} w\rightharpoonup 0\;\;\;\text {in}\;\;L^{r}(\mathbb {R}^{N}). \end{aligned}$$

(1.6)

Combining (1.5)–(1.6), we find

$$\begin{aligned} \left\{ \begin{array}{l} \lim \limits _{m\rightarrow \infty }\int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q}-|v_{m}|^{q})](|u_{m}|^{q-2}u_{m}w-|v_{m}|^{q-2}v_{m}w)\, dx= \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q-2}uw\,dx,\\ \lim \limits _{m\rightarrow \infty } \int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q}-|v_{m}|^{q})]|v_{m}|^{q-2}v_{m}w\,dx=0,\\ \lim \limits _{m\rightarrow \infty } \int _{\mathbb {R}^{N}}[\mathcal {K}_{\mu }*( |u_{m}|^{q-2}u_{m} w-|v_{m}|^{q-2}w_{m} w)]|v_{m}|^{q}\,dx=0. \end{array}\right. \nonumber \\ \end{aligned}$$

(1.7)

By Hölder’s inequality and the Hardy–Littlewood–Sobolev inequality, we have

$$\begin{aligned} \left| \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|v_{m}|^{q})|v_{m}|^{q-2}v_{m}w\,dx\right|\le & {} C(N,\mu , r)\Vert v_{m}\Vert ^{q}_{rq} \Vert |v_{m}|^{q-2}u_{m}w\Vert _{r}\nonumber \\\le & {} C\Vert |v_{m}|^{q-2}v_{m}w\Vert _{r}. \end{aligned}$$

(1.8)

On the other hand, using Lemma 2.4 we infer that $|v_{m}|^{(q-1)r}\rightharpoonup 0$ in $L^{\frac{q}{q-1}}(\mathbb {R}^{N})$, so

$$\begin{aligned} \left( \int _{\mathbb {R}^{N}}|v_{m}|^{(q-1)r} |w|^{r}\right) ^{1/r}\rightarrow 0. \end{aligned}$$

Hence, from (1.8) we obtain

$$\begin{aligned} \lim _{m\rightarrow \infty } \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|v_{m}|^{q})|v_{m}|^{q-2}v_{m}w\,dx=0. \end{aligned}$$

(1.9)

Using (1.7) and (1.9) and passing to the limit in (1.4) as $m\rightarrow \infty$, we reach the conclusion. $\square$

3 Properties Involving the Functional E Restricted to $\mathcal {N}$

In this section, we provide some properties involving the functional E restricted to the Nehari manifold $\mathcal {N}$. These properties turn out to be very useful in discussing the global existence and blow-up of solutions for initial value problem (1.1).

For $\lambda > 0$, we consider the function g defined by

$$\begin{aligned} g(\lambda )=E(\lambda u)=\frac{\lambda ^{\theta p}}{\theta p}\Vert u\Vert ^{\theta p}-\frac{\lambda ^{2q}}{2q}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx. \end{aligned}$$

(3.1)

Lemma 3.1

Let $u\in W\backslash \{0\}$ and $\theta p< 2q$. Then we have

• (1) $\lim _{\lambda \rightarrow 0^{+}}g(\lambda )=0$ and $\lim _{\lambda \rightarrow +\infty }g(\lambda )=-\infty .$

• (2) There is a unique $\lambda ^{*}=\lambda ^{*}(u)> 0$ such that $g'(\lambda ^{*})=0$.

• (3) $g(\lambda )$ is increasing on $(0,\lambda ^{*})$, decreasing on $(\lambda ^{*}, +\infty )$ and attains the maximum at $\lambda ^{*}$.

• (4) $I(\lambda u)> 0$ for $0< \lambda < \lambda ^{*}$, $I(\lambda u)< 0$ for $\lambda ^{*}< \lambda < +\infty$ and $I(\lambda ^{*}u)=0$.

Proof

For $u\in W\backslash \{0\}$, by definition of $g(\lambda )=E(\lambda u)$, it is clear that the first statement holds due to $\theta p< 2q$. Now, by differentiating $g(\lambda )$ we obtain

$$\begin{aligned} \frac{d}{d\lambda }g(\lambda )=\lambda ^{\theta p-1}\left( \Vert u\Vert ^{\theta p}-\lambda ^{2q-\theta p}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx\right) . \end{aligned}$$

(3.2)

Therefore, by taking

$$\begin{aligned} \lambda ^{*}=\lambda ^{*}(u)=\left( \frac{\Vert u\Vert ^{\theta p}}{\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx}\right) ^{1/(2q-\theta p)}, \end{aligned}$$

the second and third statements can be shown easily. In order to show the fourth statement, one can check that

$$\begin{aligned} I(\lambda u)=\lambda g'(\lambda ). \end{aligned}$$

The proof is now complete. $\square$

Lemma 3.2

Let $u\in W\backslash \{0\}$ and $\theta p< 2q$. Then

• (1) If $\Vert u\Vert < \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{1/(2q-\theta p)}$, then $I(u)> 0.$

• (2) If $I(u)\le 0$, then $\Vert u\Vert \ge \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{1/(2q-\theta p)}$.

Proof

By the definition of I, we have

$$\begin{aligned} I(u)=\Vert u\Vert ^{\theta p}-\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx \end{aligned}$$

Using the Hardy–Littlewood–Sobolev inequality, we obtain

$$\begin{aligned} I(u)\ge & {} \Vert u\Vert ^{\theta p}-C(N,\mu ,r)\Lambda _{rq}^{-2q}\Vert u\Vert ^{2q}\\= & {} \Vert u\Vert ^{ \theta p}\left( 1-C(N,\mu , r)\Lambda _{rq}^{-2q}\Vert u\Vert ^{2q-\theta p}\right) \end{aligned}$$

Hence the proof follows from the above inequality. $\square$

The next lemma shows that the stationary problem related to (1.1) admits a ground state solution.

Lemma 3.3

Assume that $\theta p < q$ holds. Then $d\ge \left( \frac{2q-\theta p}{2\theta qp}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}$ and there exists $u\in \mathcal {N}$ such that $d=E(u)$.

Proof

For any $u\in \mathcal {N}$, from Lemma 3.2 we know that $\Vert u\Vert ^{\theta p}\ge \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}$. Then by (1.6) with $I(u)=0$, it follows that

$$\begin{aligned} E(u)=\left( \frac{2q-\theta p}{2\theta qp}\right) \Vert u\Vert ^{\theta p} \ge \left( \frac{2q-\theta p}{2\theta qp}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}> 0. \end{aligned}$$

By the definition of d we obtain $d\ge \left( \frac{2q-\theta p}{2\theta qp}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}$. It remains to prove the second part of the lemma. Let $\{u_{n}\}_{n}$ be a minimizing sequence of d, i.e, $\{u_{n}\}_{n}\subset \mathcal {N}$ and $E(u_{n})\rightarrow d$ as $n\rightarrow \infty$. From (1.6) we have

$$\begin{aligned} E(u_{n})=\left( \frac{2q-\theta p}{2\theta qp}\right) \Vert u_{n}\Vert ^{\theta p}, \end{aligned}$$

which implies that $\{u_{n}\}_{n}$ is bounded in W. Going if necessary to a subsequence, we assume that

$$\begin{aligned} \left\{ \begin{array}{lll} u_{n} \rightharpoonup u &{}\text {weakly in }&{} W, \\ u_{n} \rightarrow u &{}\text {strongly in }&{} L^{\nu }(\mathbb {R}^{N}),\;\forall \nu \in [2,p^{*}_{s}), \\ u_{n} \rightarrow u &{}\text {a.e. in }&{} \mathbb {R}^{N}.\\ \end{array} \right. \end{aligned}$$

(3.3)

Now we claim that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{n}|^{q})|u_{n}|^{q}\,dx=\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx. \end{aligned}$$

(3.4)

By a direct computation we have

$$\begin{aligned}&\left| \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{n}|^{q})|u_{n}|^{q}\,dx-\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx\right| \\&\quad \le \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*\left| |u_{n}|^{q}-|u|^{q}\right| )|u_{n}|^{q}\,dx\\&\qquad +\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})\left| |u_{n}|^{q}-|u|^{q}\right| \,dx. \end{aligned}$$

Since $2< qr<p^{*}_{s}$, by using the Hardy–Littlewood–Sobolev inequality, we have

$$\begin{aligned}&\left| \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{n}|^{q})|u_{n}|^{q}\,dx-\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx\right| \\&\quad \le 2 C(N,\mu ,r)\Vert u_{n}-u\Vert _{qr} \Vert u_{n}\Vert ^{2q-1}_{qr}\\&\quad \le C_{1}\Vert u_{n}-u\Vert _{qr}\rightarrow 0 \;\text {as}\,n\rightarrow \infty . \end{aligned}$$

Since $I(u_{n})=0$, it follows from Lemma 3.2 that $\Vert u_{n}\Vert ^{\theta p}\ge \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}.$ By the definition of I we have

$$\begin{aligned} \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}\le \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u_{n}|^{q})|u_{n}|^{q}\,dx. \end{aligned}$$

So it follows from (3.4) and the above inequality that

$$\begin{aligned} 0<\left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\theta p/(2q-\theta p)}\le \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u|^{q})|u|^{q}\,dx. \end{aligned}$$

$\square$

This ensures that $u\ne 0$. Combining (3.3)–(3.4) and the weak lower semicontinuity of the norm, yield

$$\begin{aligned} E(u)\le \lim _{n\rightarrow \infty } E(u_{n})\le d, \end{aligned}$$

(3.5)

and

$$\begin{aligned} I(u)\le \lim _{n\rightarrow \infty } I(u_{n})=0. \end{aligned}$$

To finish the proof we need to show that $I(u)=0$. Indeed, arguing by contradiction, by supposing that $I(u)< 0$. Since $u\in W\backslash \{0\}$ it follows from Lemma 3.1 that there exists $\lambda ^{*}\in (0,1)$ such that $I(\lambda ^{*}u)=0$. Hence,

$$\begin{aligned}&d\le E(\lambda ^{*}u)=(\lambda ^{*})^{\theta }\left( \frac{1}{\theta p}-\frac{1}{2q}\right) \Vert u\Vert ^{\theta p}\\&\quad \le (\lambda ^{*})^{\theta }\liminf _{n\rightarrow \infty }\left( \frac{1}{\theta p}-\frac{1}{2q}\right) \Vert u_{n}\Vert ^{\theta p}<\liminf _{n\rightarrow \infty }E(u_{n})=d, \end{aligned}$$

which is absurd. Hence $I(u)=0$. Therefore, $u\in \mathcal {N}$ and the conclusion follows immediately from (3.5).

4 Local Existence

This section is devoted to the proof of Theorem (1.1). In the sequel we will use the Galerkin method.

In what follows (., .) denotes the inner product in $L^{2}(\mathbb {R}^{N})$ and $v^{+}=\max \{v,0\}$. Since W is separable and W dense in $L^{2}(\mathbb {R}^{N})$, we have a base $\mathcal {V}=\{w_{i}, i \in \mathbb {N}\}$ in W, and also in $L^{2}(\mathbb {R}^{N})$ such that $(w_{i}, w_{j})=\delta _{i,j}$, $i,j=1,2,\ldots .$ For $m\in \mathbb {N}^{*}$ we look for the approximate solution $u_{m}(x,t)=\sum \limits _{i=1}^{m}g_{im}(t)w_{i}$ satisfying the following identity :

$$\begin{aligned}&\int _{\mathbb {R}^{N}}u_{mt} w_{j}\,dx+\Vert u_{m}\Vert ^{ p(\theta -1)}\langle u_{m}, w_{j}\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q-2}u^{+}_{m}w_{j}\,dx,\;j=1,2,\ldots ,m, \end{aligned}$$

(4.1)

and

$$\begin{aligned} u_{m}(0)=u_{0m}, \end{aligned}$$

(4.2)

where

$$\begin{aligned} \langle u,w_{j}\rangle= & {} \int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(w_{j}(x)-w_{j}(y))}{|x-y|^{N+sp}}\,dxdy\\&\quad +\int _{\mathbb {R}^{N}} V|u|^{p-2}uw_{j}\,dx. \end{aligned}$$

Since $\mathcal {V}=\{w_{i}, i \in \mathbb {N}\}$ is dense in W and $u_{0}\in W$, there exists $\{\eta _{im}, i=1,2,\ldots ,m\}$ such that

$$\begin{aligned} u_{m}(0)=\sum _{i=1}^{m}\eta _{im}w_{i}\rightarrow u_{0}\;\;\;\text {strongly in }\,W. \end{aligned}$$

(4.3)

Then (4.1)–(4.2) is reduced to the following initial value problem for a system of nonlinear differential equations on $g_{jm}(t)$ :

$$\begin{aligned} \left\{ \begin{array}{c} g'_{jm}(t)=G_{j}(g), \;\;j=1,\ldots , m, \\ g_{jm}(0)=\eta _{jm}, \;\; j=1,\ldots ,m, \end{array} \right. \end{aligned}$$

(4.4)

where

$$\begin{aligned} G_{j}(g)=-\Vert u_{m}\Vert ^{p(\theta -1)}\langle u_{m}, w_{j}\rangle +\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q-2}u_{m}w_{j}\,dx. \end{aligned}$$

By the Picard iteration method, there exists $t_{0,m}>0$ depending on $|\eta _{jm}|$ such that problem (4.4) admits a unique solution $g_{jm}\in C^{1}([0,t_{0,m}]).$

Multiplying (4.1) by $g_{jm}(t)$ and summing over $j=1,2,\ldots ,m$, yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u_{m}(t)\Vert ^{2}_{2}+\Vert u_{m}(t)\Vert ^{\theta p}=\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q}\,dx, \;\;\forall t\in [0,t_{0,m}]. \end{aligned}$$

(4.5)

By using the Hardy–Littlewood–Sobolev inequality, we obtain

$$\begin{aligned} \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q}\,dx\le C(N,\mu , r)\Vert u_{n}\Vert ^{2q}_{rq}. \end{aligned}$$

From (1.11) we have $2< qr< p^{*}_{s}$, by using the interpolation inequality and the continuous embedding $W\hookrightarrow L^{p^{*}_{s}} (\mathbb {R}^{N})$, we find

$$\begin{aligned} \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q}\,dx\le \Lambda _{p^{*}_{s}}^{-2q(1-\gamma )}C(N,\mu , r)\Vert u_{n}\Vert ^{2q\gamma }_{2}\Vert u_{m}\Vert ^{2q(1-\gamma )},\nonumber \\ \end{aligned}$$

(4.6)

where $\gamma \in (0,1)$ satisfies

$$\begin{aligned} \frac{1}{qr}=\frac{\gamma }{2}+\frac{1-\gamma }{p^{*}_{s}}. \end{aligned}$$

Since $2<qr<2q< \min \left\{ \theta p+\frac{4p^{*}_{s}-2r\theta p}{rp^{*}_{s}}, \frac{(2-r)p^{*}_{s}}{2}\right\}$, we have

$$\begin{aligned} 2q(1-\gamma )< \theta p, \end{aligned}$$

and

$$\begin{aligned} \alpha =\frac{\theta pq\gamma }{\theta p-2q(1-\gamma )}>1. \end{aligned}$$

Using the Young inequality in (4.6), for any $\varepsilon \in (0,1)$ we obtain

$$\begin{aligned} \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q}\,dx\le C(\varepsilon )\left( \Vert u_{m}(t)\Vert _{2}^{2}\right) ^{\alpha }+\varepsilon \Vert u_{m}(t)\Vert ^{\theta p}, \end{aligned}$$

(4.7)

Combining (4.5) and (4.7), we get

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\Vert u_{m}(t)\Vert ^{2}_{2}+(1-\varepsilon )\Vert u_{m}(t)\Vert ^{\theta p}\le C(\varepsilon )\left( \Vert u_{m}(t)\Vert _{2}^{2}\right) ^{\alpha }, \;\;\forall t\in [0,t_{0,m}]. \end{aligned}$$

Taking $\varepsilon =\frac{1}{2}$, yields

$$\begin{aligned} \frac{d}{dt}\Vert u_{m}(t)\Vert ^{2}_{2}\le 2C_{1}\left( \Vert u_{m}(t)\Vert _{2}^{2}\right) ^{\alpha }, \;\;\forall t\in [0,t_{0,m}]. \end{aligned}$$

(4.8)

Since $\alpha >1$, according to (4.8), we obtain

$$\begin{aligned} \Vert u_{m}(t)\Vert ^{2}_{2}\le \left( C_{2}^{1-\alpha }-2(\alpha -1)C_{1}t\right) ^{\frac{1}{1-\alpha }}. \end{aligned}$$

(4.9)

only if $t< T^{*}=\frac{C^{1-\alpha }_{2}}{2(\alpha -1)C_{1}}$, where $C_{2}=\sup _{m}\int _{\mathbb {R}^{N}}u^{2}_{m}(x,0)\,dx\in (0, +\infty )$. It follows that

$$\begin{aligned} \Vert u_{m}(t)\Vert ^{2}_{2}\le 2^{\frac{1}{\gamma -1}} C_{2}, \;\;\forall t\le \min \left\{ t_{0,m}, T^{*}/2\right\} . \end{aligned}$$

(4.10)

Now, we claim that (4.10) holds for all $t\in [0, T^{*}/2]$. Indeed, if $T^{*}/2\le t_{0,m}$, then there is nothing to prove. Otherwise, if $t_{0,m}< T^{*}/2$, then $\Vert u(t_{0,m})\Vert ^{2}_{2}\le 2^{\frac{1}{\gamma -1}} C_{2}$. Thus, we can replace $u_{0m}$ in (4.2) by $u_{m}(x,t_{0,m})$ and extend the solution to the interval $[0,T^{*}/2]$ by repeating the above process. Thus, we obtain

$$\begin{aligned} \Vert u_{m}(t)\Vert ^{2}_{2}\le 2^{\frac{1}{\gamma -1}} C_{2}, \;\;\forall t\in [0,T] \;\;\left( T=T^{*}/2\right) . \end{aligned}$$

(4.11)

Now, multiplying the $j^{th}$ equations in (4.1) by $g'_{jm}(t)$ and summing over j from 1 to m, afterward, integrating over (0, t) yields

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds+E(u_{m}(t))=E(u_{0m}), \;\;t\in [0,T]. \end{aligned}$$

(4.12)

From (4.3) and the continuity of the functional E on W, we have

$$\begin{aligned} |E(u_{0m})|\le c, \;\;\forall m\in \mathbb {N}^{*}. \end{aligned}$$

(4.13)

Remembering that

$$\begin{aligned} E(u_{m}(t))=\frac{1}{\theta p}\Vert u_{m}(t)\Vert ^{\theta p}-\frac{1}{2q}\int _{\mathbb {R}^{N}}\mathcal {K}_{\mu }*(|u^{+}_{m} |^{q})|u^{+}_{m}|^{q}\,dx. \end{aligned}$$

Thus, combining (4.7), (4.11), (4.12) and (4.13) we get

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds+\left( \frac{1}{\theta p}-\varepsilon \right) \Vert u_{m}(t)\Vert ^{\theta p}\le c, \;\;\forall t\in [0,T]. \end{aligned}$$

Taking $\varepsilon =\frac{1}{2\theta p}$, we conclude that

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds\le c_{1}, \;\;\forall t\in [0,T], \end{aligned}$$

(4.14)

and

$$\begin{aligned} \Vert u_{m}(t)\Vert \le c_{2}, \;\;\forall t\in [0,T]. \end{aligned}$$

(4.15)

Then, by (4.14) and (4.15), there exist u and a subsequence of $\{u_{m}\} _{m}$, still denoted by $\{u_{m}\} _{m}$ such that, as $m\rightarrow \infty$

$$\begin{aligned} \left\{ \begin{array}{lcl} u_{m}\rightharpoonup ^{*} u &{} \text {in}&{} L^{\infty }(0,T;W)),\\ u_{mt}\rightharpoonup u_{t} &{}\text {in} &{} L^{2}(0,T;L^{2}(\mathbb {R}^{N})), \end{array}\right. \end{aligned}$$

(4.16)

Then it follows from Aubin-Lions compactness theorem, see [39, Corollary 4] that

$$\begin{aligned} u_{m}\rightarrow u \;\; \text {in} \;\;\;C([0,T], L^{\nu }(\mathbb {R}^{N})), \;\;\forall \nu \in [2, p^{*}_{s}). \end{aligned}$$

(4.17)

This implies

$$\begin{aligned} u_{m}(x,t) \rightarrow u(x,t) \;\;\text {a.e. in }\; \mathbb {R}^{N}, \;\forall t\ge 0. \end{aligned}$$

(4.18)

Now, by using [40, proposition 1.3] we prove that $\{\Vert u_{m}(t)\Vert ^{p(\theta -1)}\}_{m}$ is relatively compact in $L^{1}(0,T)$. Indeed, by (4.15) we have $\Vert u_{m}(t)\Vert ^{p (\theta -1)}\le c$, for all m and t. This implies that $\int _{0}^{T}\Vert u_{m}(t)\Vert ^{p(\theta -1)}\,dt\le cT$ for all m. On the other hand, for any $\varepsilon >0$, there exits $\delta =\frac{\varepsilon }{c}$ such that for any measurable subset A with $|A|<\delta$, there holds

$$\begin{aligned} \int _{A}\Vert u_{m}(t)\Vert ^{p (\theta -1)}\,dt\le c|A|<\varepsilon . \end{aligned}$$

Consequently, $\{\Vert u_{m}(t)\Vert ^{p (\theta -1)}\}_{m}$ is relatively compact in $L^{1}(0,T)$. Therefore, there exist $\beta (t)\in L^{1}(0,T)$ and a subsequence of $\{\Vert u_{m}(t)\Vert ^{p (\theta -1)}\}_{m}$, still denoted by $\{\Vert u_{m}(t)\Vert ^{p(\theta -1)}\}_{m}$ such that

$$\begin{aligned} \Vert u_{m}(t)\Vert ^{p(\theta -1)}\rightarrow \beta (t), \;\;\text {a.e.}\; t\in [0,T]. \end{aligned}$$

(4.19)

From (4.16) and [41, Lemma 3.1.7], we infer that

$$\begin{aligned} u_{m}(0)\rightarrow u(0)\;\;\text {weakly in}\; L^{2}(\mathbb {R}^{N}). \end{aligned}$$

(4.20)

On the other hand, from (4.3) we have $u_{m}(0)\rightarrow u_{0}$ strongly in W. Thus, this combined with (4.20) yields

$$\begin{aligned} u(0)=u_{0}. \end{aligned}$$

(4.21)

In the next step, we show that the equality (1) in Definition 1.1 holds.

First, we claim that

$$\begin{aligned} u^{+}_{m}\rightarrow u^{+} \;\; \text {in} \;\;\;C([0,T], L^{\nu }(\mathbb {R}^{N})), \;\;\forall \nu \in [2, p^{*}_{s}). \end{aligned}$$

(4.22)

To this end, for any $\nu \in [2, p^{*}_{s})$ we have

$$\begin{aligned}&\int _{\mathbb {R}^{N}}|u^{+}_{m}-u^{+}|^{\nu }\,dx= \int _{\mathbb {R}^{N}}\left| \frac{|u_{m}|-|u|}{2}-\frac{(u_{m}-u)}{2}\right| ^{\nu }\,dx\\&\quad \le \int _{\mathbb {R}^{N}} |u_{m}-u|^{\nu }\,dx\rightarrow 0 \;\text {as}\;m\rightarrow \infty . \end{aligned}$$

Therefore, the claim holds. Combining (3.4) with (4.22), yields

$$\begin{aligned} \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}_{m})^{q})(u^{+}_{m})^{q}\,dx\rightarrow \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+})^{q})(u^{+})^{q}\,dx, \;\;\text {as}\;m\rightarrow \infty .\nonumber \\ \end{aligned}$$

(4.23)

On the other hand, by (4.22) and Lemma 2.6 we get

$$\begin{aligned}&\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}_{m})^{q-1})(u^{+}_{m})^{q-1}w\,dx \nonumber \\&\quad \rightarrow \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+})^{q-1})(u^{+})^{q-1}w\,dx, \;\;\text {as}\;m\rightarrow \infty , \;\;\;\forall w\in L^{rq}(\mathbb {R}^{N}).\nonumber \\ \end{aligned}$$

(4.24)

Now, let us consider the linear operator $L :W\rightarrow W'$ defined by

$$\begin{aligned} \langle L(u), v\rangle =\int _{\mathbb {R}^{2N}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}\,dxdy. \end{aligned}$$

From here on we set $p'=\frac{p-1}{p}$ is the Hölder conjugate of p. In view of (4.15) we derive

$$\begin{aligned} \left\{ \frac{|u_{m}(x, t)-u_{m}(y, t)|^{p-2}(u_{m}(x, t)-u_{m}(y, t))}{|x-y|^{\frac{N+sp}{p'}}}\right\} _{m} \;\;\text {is bounded in}\;L^{p'}(\mathbb {R}^{2N}), \;\forall t\in [0,T], \end{aligned}$$

and

$$\begin{aligned}&\frac{|u_{m}(x, t)-u_{m}(y, t)|^{p-2}(u_{m}(x, t)-u_{m}(y, t))}{|x-y|^{\frac{N+sp}{p'}}}\\&\quad \rightarrow \frac{|u(x, t)-u(y, t)|^{p-2}(u(x, t)-u(y, t))}{|x-y|^{\frac{N+sp}{p'}}}\;\;\text {a.e. in}\;\mathbb {R}^{2N}, \;\;\forall t\ge 0. \end{aligned}$$

Invoking Lemma (2.4), we deduce

$$\begin{aligned}&\frac{|u_{m}(x, t)-u_{m}(y, t)|^{p-2}(u_{m}(x, t)-u_{m}(y, t))}{|x-y|^{\frac{N+sp}{p'}}} \nonumber \\&\quad \rightarrow \frac{|u(x, t)-u(y, t)|^{p-2}(u(x, t)-u(y, t))}{|x-y|^{\frac{N+sp}{p'}}} \;\;\text {weakly in}\;L^{p'}(\mathbb {R}^{2N}). \end{aligned}$$

(4.25)

Thus, for any $v\in W$ we have

$$\begin{aligned} \frac{v(x)-v(y)}{|x-y|^{\frac{N+sp}{p}}}\in L^{p}(\mathbb {R}^{2N}), \end{aligned}$$

which implies, as $m\rightarrow \infty$

$$\begin{aligned} \langle L(u_{m}(t)),v\rangle \rightarrow \langle L(u(t)),v\rangle , \;\;\forall v\in W, \;\;\forall t\in [0,T]. \end{aligned}$$

(4.26)

Similarly, as $\{V^{\frac{p-1}{p}}(x)|u_{m}|^{p-2}u_{m}\}_{m}$ is bounded in $L^{p'}(\mathbb {R}^{N})$ and

$$\begin{aligned} V^{\frac{p-1}{p}}(x)|u_{m}|^{p-2}u_{m}\rightarrow V^{\frac{p-1}{p}}(x)|u|^{p-2}u\;\;\;\text {a.e. in}\;\mathbb {R}^{N}, \,t\ge 0, \end{aligned}$$

we obtain

$$\begin{aligned} V^{\frac{p-1}{p}}(x)|u_{m}|^{p-2}u_{m}\rightarrow V^{\frac{p-1}{p}}(x)|u|^{p-2}u\;\;\;\text {weakly in}\;\;L^{p'}(\mathbb {R}^{N}), \end{aligned}$$

For any $w\in W$, we have

$$\begin{aligned} V^{\frac{1}{p}}w\in L^{p}(\mathbb {R}^{N}). \end{aligned}$$

Hence, as $m\rightarrow \infty$

$$\begin{aligned} \int _{\mathbb {R}^{N}}V(x)|u_{m}|^{p-2}u_{m}w\,dx\rightarrow \int _{\mathbb {R}^{N}}V(x)|u|^{p-2}uw\,dx, \;\;\forall w\in W. \end{aligned}$$

(4.27)

Combining (4.26) with (4.19) and using the Lebesgue dominated convergence theorem, yields

$$\begin{aligned} \int _{0}^{T}\Vert u_{m}(t)\Vert ^{\theta (p-1)}\langle L(u_{m}(t)),v\rangle \,dt\rightarrow \int _{0}^{T}\beta (t)\langle L(u(t)),v\rangle \,dt,\;\;\;\forall v\in W.\nonumber \\ \end{aligned}$$

(4.28)

Integrating (4.1) with respect to t, we obtain

$$\begin{aligned}&\int _{0}^{T}\int _{\mathbb {R}^{N}}u_{mt} w_{j}\,dxdt+\int _{0}^{T}\Vert u_{m}\Vert ^{ p(\theta -1)}\langle u_{m}, w_{j}\rangle \,dt\nonumber \\&\quad =\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}_{m}|^{q})|u^{+}_{m}|^{q-2}u^{+}_{m}w_{j}\,dxdt,\;j=1,2,\ldots ,m. \end{aligned}$$

(4.29)

In view of (4.17), (4.24) and (4.28), for j fixed, we can pass to the limit in (4.29) to get

$$\begin{aligned}&\int _{0}^{T}\int _{\mathbb {R}^{N}}u_{t} w_{j}\,dxdt+\int _{0}^{T}\beta (t)\langle u, w_{j}\rangle \,dt \nonumber \\&\quad =\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*|u^{+}|^{q})|u^{+}|^{q-2}u^{+}w_{j}\,dxdt,\;j=1,2,\ldots ,m. \end{aligned}$$

(4.30)

Since $\mathcal {V}=\{w_{i},\,i\in \mathbb {N}\}$ is dense in W, we infer from (4.30) that

$$\begin{aligned}&\int _{0}^{T}\int _{\mathbb {R}^{N}}u_{t} v\,dxdt+\int _{0}^{T}\beta (t)\langle u, v\rangle \,dt\nonumber \\&\quad =\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+})^{q})(u^{+})^{q-1}v\,dxdt,\;\forall v\in W, \end{aligned}$$

(4.31)

which implies

$$\begin{aligned}&\int _{\mathbb {R}^{N}}u_{t}(t) v\,dx+\beta (t)\langle u(t), v\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}(t))^{q})(u^{+}(t))^{q-1}v\,dx,\;\;\text {a.e.}\;t\in [0,T],\;\forall v\in W. \end{aligned}$$

(4.32)

If $\beta (t)=0$. Then by (4.19), we deduce that the equality (1) in the Definition 1.1 clearly holds.

Assume $\beta (t) >0$ for all $t\in [0,T]$. Taking $u=v$ in (4.31) and using [33, proposition 2.1], we get

$$\begin{aligned} \Vert u(T)\Vert ^{2}_{2}-\Vert u_{0}\Vert ^{2}_{2}+ \int _{0}^{T}\beta (t)\Vert u(t)\Vert ^{p} \,dt=\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+})^{q})(u^{+})^{q}\,dxdt.\nonumber \\ \end{aligned}$$

(4.33)

On the other hand, multiplying both sides of equations in (4.1) by $g_{jm}(t)$, and summing with respect to i, afterward integrating over (0, T) yields

$$\begin{aligned}&\Vert u_{m}(T)\Vert ^{2}_{2}-\Vert u_{0m}\Vert ^{2}_{2}+ \int _{0}^{T}\Vert u_{m}\Vert ^{(\theta -1)p}\langle u_{m}, u_{m}\rangle \,dt\nonumber \\&\quad =\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u_{m}^{+})^{q})(u_{m}^{+})^{q}\,dxdt. \end{aligned}$$

(4.34)

Combining (4.33) with (4.34), and using (4.17) we obtain

$$\begin{aligned}&\lim _{m\rightarrow } \int _{0}^{T}\Vert u_{m}\Vert ^{p(\theta -1)}\langle u_{m}, u_{m}\rangle \,dt\\&\quad =-\Vert u(T)\Vert ^{2}_{2}-\Vert u_{0}\Vert ^{2}_{2}+\int _{0}^{T}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+})^{q})(u^{+})^{q}\,dxdt\\&\quad = \int _{0}^{T}\beta (t)\Vert u(t)\Vert ^{p} \,dt\\&\quad = \lim _{m\rightarrow \infty }\int _{0}^{T}\Vert u_{m}\Vert ^{p(\theta -1)}\langle u_{m}, u\rangle \,dt. \end{aligned}$$

Thus, we deduce that

$$\begin{aligned} \lim _{m\rightarrow \infty } \int _{0}^{T}\Vert u_{m}\Vert ^{p(\theta -1)}\langle u_{m}, u_{m}-u\rangle \,dt=0. \end{aligned}$$

(4.35)

Hence, there exists a subsequence, still denoted by $\{u_{m}\}$ such that for a.e. $t\in [0,T]$

$$\begin{aligned} \lim _{m\rightarrow \infty } \left( \Vert u_{m}(t)\Vert ^{p(\theta -1)}\langle u_{m}(t), u_{m}(t)-u(t)\rangle \right) =0. \end{aligned}$$

By (4.19), as $m\rightarrow \infty$ we know that

$$\begin{aligned} \Vert u_{m}(t)\Vert ^{p(\theta -1)}\rightarrow \beta (t)>0, \;\;\text {a.e.}\; t\in [0,T], \end{aligned}$$

(4.36)

which implies

$$\begin{aligned} \lim _{m\rightarrow \infty } \Vert u_{m}(t)\Vert ^{p}=\Vert u(t)\Vert ^{p}\;\;\;\text {a.e.}\;t\in [0,T]. \end{aligned}$$

Obviously, $\lim \limits _{m\rightarrow \infty } \Vert u_{m}(t)\Vert ^{p(\theta -1)}= \Vert u(t)\Vert ^{p(\theta -1)}$ a.e. $t\in [0,T]$. Therefore, we obtain

$$\begin{aligned} \beta (t) =\Vert u(t)\Vert ^{p(\theta -1)}\;\;\text { a.e. }\;t\in [0,T]. \end{aligned}$$

(4.37)

Inserting (4.37) in (4.32), yields

$$\begin{aligned}&\int _{\mathbb {R}^{N}}u_{t}(t) v\,dx+\Vert u(t)\Vert ^{p(\theta -1)}\langle u(t), v\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}(t))^{q})(u^{+}(t))^{q-1}v\,dx,\;\;\text {a.e.}\;t\in [0,T],\;\forall v\in W. \end{aligned}$$

(4.38)

It remains to show that inequality (1.12) holds. From (3.5), (4.16), (4.24), (4.19) and $u_{0m}\rightarrow u_{0}$ in W, we infer

$$\begin{aligned}&\int _{0}^{t}\Vert u_{s}(s)\Vert ^{2}_{2}\,ds+E(u(t))\le \liminf _{m\rightarrow \infty }\left( \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds+E(u_{m}(t))\right) \nonumber \\&\quad =\lim _{m\rightarrow \infty }E(u_{0m})=E(u_{0}). \end{aligned}$$

Corollary 4.1

If $u_{0}(x)\ge 0$ a.e. in $\mathbb {R}^{N}$. Then the function $u(x,t)\ge 0$ a.e. in $\mathbb {R}^{N}$ and for any $t\in [0,T].$

Proof

Let u be a weak solution to initial value problem (1.1). Clearly

$$\begin{aligned} u^{-}=\max \{-u, 0\}\in L^{\infty }(0,T; W). \end{aligned}$$

Taking $v=-u^{-}$ in (4.38), we obtain

$$\begin{aligned}&-\int _{\mathbb {R}^{N}}u_{t}(t)u^{-}(t) \,dx+\Vert u(t)\Vert ^{p(\theta -1)}\langle u(t), -u^{-}(t)\rangle \nonumber \\&\quad =-\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}(t))^{q})(u^{+}(t))^{q-1}u^{-}(t)\,dx, \end{aligned}$$

(4.39)

Observe that for a.e. $x,y\in \mathbb {R}^{N}$,

$$\begin{aligned}&|u(x)-u(y)|^{p-2}(u(x)-u(y))(-u^{-}(x)+u^{-}(y))\\&\quad =|u(x)-u(y)|^{p-2}(u^{-}(x)-u^{-}(y))^{2}\\&\quad \quad +|u(x)-u(y)|^{p-2}u^{+}(y)u^{-}(x)+|u(x)-u(y)|^{p-2}u^{-}(y)u^{+}(x)\\&\quad \ge |u^{-}(x)-u^{-}(y)|^{p}, \end{aligned}$$

and

$$\begin{aligned} -V(x)|u(x)|^{p-2}u(x)u^{-}(x)=V(x)|u(x)|^{p-2}(u^{-}(x))^{2}\ge V(x)|u^{-}|^{p}. \end{aligned}$$

Furthermore, from the definition of $u^{+}$ and $u^{-}$, we get

$$\begin{aligned} \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*(u^{+}(t))^{q})(u^{+}(t))^{q-1}u^{-}(t)\,dx=0. \end{aligned}$$

Combining these facts with (4.39), yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _{\mathbb {R}^{N}}(u^{-}(t))^{2}\,dx+\Vert u^{-}(t)\Vert ^{p\theta }\le 0, \end{aligned}$$

which implies

$$\begin{aligned} \int _{\mathbb {R}^{N}}(u^{-}(t))^{2}\,dx\le \int _{\mathbb {R}^{N}}(u_{0}^{-})^{2}\,dx, \;\;\forall t\in [0,T]. \end{aligned}$$

Since $u_{0}(x)\ge 0$ a.e. in $\mathbb {R}^{N}$ we get $u^{-} (t)=0$ a.e. in $\mathbb {R}^{N}$ and for any $t\in [0,T]$. Hence, $u(x,t)\ge 0$ a.e. in $\mathbb {R}^{N}$ and for any $t\in [0,T]$. $\square$

5 Global Existence and Asymptotic Behavior

This section is concerned with the proof of the existence of global weak solutions to initial value problem (1.1). Moreover, we show the asymptotic behavior of these solutions as $t\rightarrow +\infty$. In doing so, we introduce the potential well set

$$\begin{aligned} \mathcal {W}=\left\{ u \in W\,: E(u)< d, \,I(u)> 0\right\} \cup \left\{ 0\right\} . \end{aligned}$$

(5.1)

Similarly to the previous section, we will employ the Galerkin method.

Proof

(Theorem 1.2) In what follows we only need to consider the case where $u_{0}\in \mathcal {W}\backslash \{0\}$. Otherwise if we assume $u_{0}=0$, then initial value problem (1.1) admits a global solution $u=0$ and there is nothing to prove. From (4.12) remembering that

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds+E(u_{m}(t))=E(u_{0m}), \;\;t\in [0,T_{m}), \end{aligned}$$

(5.2)

where $T_{m}$ is the maximal existence time of solution $u_{m}$ to initial value problem (4.1)–(4.2). Since we are assuming that $u_{0}\in \mathcal {W}\backslash \{0\}$, that is

$$\begin{aligned} E(u_{0})< d \quad \text {and }\quad I(u_{0})>0. \end{aligned}$$

It is follows from (4.3) that

$$\begin{aligned} E(u_{0m})< d \quad \text {and }\quad I(u_{0m})>0, \end{aligned}$$

(5.3)

for m large enough. This combined with (5.2), yields

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds+E(u_{m}(t))<d, \;\;t\in [0,T_{m}). \end{aligned}$$

(5.4)

for m large enough. Next we prove $u_{m}\in \mathcal {W}$ for sufficiently large m and $T_{m}=+\infty$. Assume that $u_{m}(t)\notin \mathcal {W}$ and let $t_{0}$ be the smallest time for which $u_{m}(t_{0})\notin \mathcal {W}$. By the continuity of $u_{m}(t)$ we obtain that $u_{m}(x, t_{0})\in \partial \mathcal {W}$, i.e.

$$\begin{aligned} E(u_{m}(t_{0}))=d. \end{aligned}$$

(5.5)

or

$$\begin{aligned} u_{m}(t_{0})\ne 0\quad \text {and}\quad I(u_{m}(t_{0}))=0. \end{aligned}$$

(5.6)

Obviously, the case (5.5) could not occur due to (5.4), while if (5.6) holds, then by definition of d in (1.7) we infer that $E(u_{m}(t_{0}))\ge d$. This is also impossible, since it contradicts (5.4).

In conclusion $u_{m}(t)\in \mathcal {W}$ for all $t\in [0, T_{m})$ and sufficiently large m, so that

$$\begin{aligned} E(u_{m}(t))< d \quad \text {and }\quad I(u_{m}(t))>0. \;\;\;\forall t\in [0,T_{m}). \end{aligned}$$

(5.7)

This combined with (1.6), for m large enough we get

$$\begin{aligned} d> E(u_{m}(t))=\frac{2q-\theta p}{2q\theta p}\Vert u_{m}(t)\Vert ^{\theta p}+\frac{1}{2q}I(u_{m}(t))>\frac{2q-\theta p}{2q\theta p}\Vert u_{m}(t)\Vert ^{\theta p}, \end{aligned}$$

(5.8)

which gives

$$\begin{aligned} \Vert u_{m}(t)\Vert <\left( \frac{2q\theta p}{2q-\theta p}\right) ^{\frac{1}{\theta p}}, \;\;\forall t\in [0, T_{m}). \end{aligned}$$

(5.9)

Furthermore, from (5.4) we observe that

$$\begin{aligned} \int _{0}^{t}\Vert u_{ms}(s)\Vert ^{2}_{2}\,ds<d, \;\;\forall t\in [0, T_{m}). \end{aligned}$$

(5.10)

In view of (5.9), we infer that $T_{m}=+\infty$. Consequently, by (5.9)–(5.10), there exist u and a subsequence of $\{u_{m}\} _{m}$, still denoted by $\{u_{m}\} _{m}$ such that, as $m\rightarrow \infty$

$$\begin{aligned} \left\{ \begin{array}{lcl} u_{m}\rightharpoonup ^{*} u &{} \text {in}&{} L^{\infty }(0,+\infty ; W)),\\ u_{mt}\rightharpoonup u_{t} &{}\text {in} &{} L^{2}(0,+\infty ;L^{2}(\mathbb {R}^{N})), \end{array}\right. \end{aligned}$$

(5.11)

Hence, in a similar manner to the previous section, applying (5.11) we conclude that initial value problem (1.1) admits a global positive solution, that is

$$\begin{aligned}&\int _{\mathbb {R}^{N}}u_{t}(t) v\,dx+\Vert u(t)\Vert ^{p(\theta -1)}\langle u(t), v\rangle \nonumber \\&\quad =\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}(t)u^{q-1}(t)v\,dx,\;\;\text {a.e.}\;t\in (0,+\infty ),\;\forall v\in W. \end{aligned}$$

(5.12)

Thus, this completes the proof of Theorem (1.2). $\square$

We give now the proof of Corollary 1.3.

Proof

(Corollary 1.3) We divided the proof into two cases :

Case 1. $E(u_{0})< d$ and $I(u_{0}) =0$ :

In view of Remark 2 this case can not occurs.

Case 2. $E(u_{0})= d$ and $I(u_{0}) \ge 0$ :

For $n=2,3,\ldots$ we let $\epsilon _{n}=1-\frac{1}{n}$, $u_{0n}=\epsilon _{n}u_{0}$ and consider the following problem

$$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+\Vert u\Vert ^{p(\theta -1)}[(-\Delta )^{s}_{p}u+V(x)|u|^{p-2}u]=(\mathcal {K}_{\mu }*|u|^{q})|u|^{q-2} u&{} \text {in}\ &{} \mathbb {R}^{N}\times (0, +\infty ) , \\ u(x,0)=u_{0n}(x), &{} \text {in} &{}\mathbb {R}^{N} , \end{array}\right. \nonumber \\ \end{aligned}$$

(5.13)

According to the definition of I we have

$$\begin{aligned} I(u_{0n})= & {} \epsilon ^{\theta p}_{n}\Vert u_{0}\Vert ^{\theta p}-\epsilon ^{2q}_{n}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}_{0})u^{q}_{0} \,dx\\= & {} \epsilon _{n}^{\theta p} I(u_{0})+\epsilon ^{2q}_{n}(\epsilon ^{\theta p-2q}_{n}-1)\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}_{0})u^{q}_{0} \,dx \end{aligned}$$

In view of $I(u_{0})\ge 0$ and $0< \epsilon _{n} < 1$ it follows that

$$\begin{aligned} I(u_{0n})> 0 \end{aligned}$$

(5.14)

Moreover according to the definition of E we know

$$\begin{aligned} \frac{d}{d\epsilon _{n}}E(\epsilon _{n}u_{0})=\frac{1}{\epsilon _{n}} I(u_{0n})> 0 \end{aligned}$$

Hence, this shows that $\epsilon _{n}\mapsto E(\epsilon _{n}u_{0})$ is strictly increasing on (0, 1), from where it follows that

$$\begin{aligned} E(u_{0n})= E(\epsilon _{n}u_{0})< E(u_{0})=d. \end{aligned}$$

(5.15)

On the other hand, it is easy to see that

$$\begin{aligned} u_{0n}\rightarrow u_{0}\;\;\;\text { in }\;\; W, \;\text { as}\; n\rightarrow \infty . \end{aligned}$$

(5.16)

Combining (5.14)–(5.16) and by a similar way to the proof of Theorem 1.2, we obtain the existence of a global solution $u_{n}$ for initial value problem (5.13) satisfies that

$$\begin{aligned} \Vert u_{n}(t)\Vert <\left( \frac{2q\theta p}{2q-\theta p}\right) ^{\frac{1}{\theta p}}, \;\;\forall t\in [0, T_{n}), \end{aligned}$$

(5.17)

and

$$\begin{aligned} \int _{0}^{t}\Vert u_{ns}(s)\Vert ^{2}_{2}\,ds<d, \;\;\forall t\in [0, T_{n}), \end{aligned}$$

(5.18)

for every $n\ge 2$. Therefore the reminder of the proof is quite similar to the proof of Theorem 1.1.

$\square$

Now we turn our attention to the asymptotic behavior of global solutions as $t\rightarrow +\infty$. To this aim, we need to recall the following Lemma which due to Martinez [42].

Lemma 5.1

Let $f :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ be a nonincreasing function and $\sigma$ is a nonnegative constant such that

$$\begin{aligned} \int _{t}^{+\infty }f^{1+\sigma }(s)\,ds\le \frac{1}{\omega }f^{\sigma }(0)f(t), \;\; \forall t\ge 0. \end{aligned}$$

then we have

• (1) $f(t)\le f(0)e^{1-\omega t}$, for all $t\ge 0$, whenever $\sigma =0$.

• (2) $f(t)\le f(0)\left( \frac{1+\sigma }{1+\omega \sigma t}\right) ^{1/\sigma }$, for all $t\ge 0$, whenever $\sigma > 0$.

Moreover, we recall the fractional Gagliardo-Nirenberg interpolation inequality that can be found in [43] : if $\Omega$ is a standard domain (i.e. $\mathbb {R}^{N}$, a half-space or it is bounded with Lipschitz boundary) then

$$\begin{aligned}&\Vert v\Vert _{W^{\tau ,p}(\Omega )}\le C \Vert v\Vert ^{\eta }_{W^{s_{1},p_{1}}(\Omega )}\Vert v\Vert ^{1-\eta }_{W^{s_{2},p_{2}}(\Omega )}, \;\;\eta \in (0,1),\nonumber \\&\tau =\eta s_{1}+(1-\eta )s_{2}, \;\;\frac{1}{p}=\frac{\eta }{p_{1}}+\frac{1-\eta }{p_{2}}. \end{aligned}$$

(5.19)

as long as it fails that : $s_{2}$ is an integer $\ge 1$ and $p_{2} =1$ and $s_{1}-s_{2}\le 1-\frac{1}{p_{1}}.$

Proof

(Theorem 1.4) Multiplying (4.1) by $g_{jm}$ and summing over $j=1,2,\ldots ,m$ gives

$$\begin{aligned} -\frac{1}{2}\frac{d}{dt}\int _{\mathbb {R}^{N}}u^{2}_{m}(t)\,dx=I(u_{m}(t)). \end{aligned}$$

(5.20)

In view of the proof of Lemma 3.2, we know that

$$\begin{aligned} I(u_{m}(t))\ge & {} \Vert u_{m}(t)\Vert ^{ \theta p}\left( 1-C(N,\mu , r)\Lambda _{qr}^{-2q}\Vert u_{m}(t)\Vert ^{2q-\theta p}\right) , \;\;\forall t\ge 0. \end{aligned}$$

On the other hand, from (1.17) for m large enough

$$\begin{aligned} E(u_{0m})< \left( \frac{2q-\theta p}{2q\theta p}\right) \left( \frac{\Lambda _{rq}^{2q}}{C(N,\mu ,r)}\right) ^{\frac{\theta p}{2q-\theta p}}. \end{aligned}$$

(5.21)

Combining (5.2) with (5.8) yields

$$\begin{aligned} \left( \frac{2q\theta p}{2q-\theta p}E(u_{0m})\right) ^{\frac{1}{\theta p}}> \Vert u_{m}(t)\Vert , \;\;\;\forall t\ge 0, \end{aligned}$$

which implies that

$$\begin{aligned} I(u_{m}(t))\ge & {} \Vert u_{m}(t)\Vert ^{ \theta p}\left( 1-C(N,\mu , r)\Lambda _{rq}^{-2q}\left( \frac{2q\theta p}{2q-\theta p}E(u_{0m})\right) ^{\frac{2q-\theta p}{\theta p}}\right) , \;\;\forall t\ge 0.\nonumber \\ \end{aligned}$$

(5.22)

Integrating (5.20) over (t, T), we infer from the continuous embedding $W\hookrightarrow L^{2}(\mathbb {R}^{N})$ and (5.22) that

$$\begin{aligned} \int _{t}^{T}\Vert u_{m}(t)\Vert _{2}^{\theta p}\,dt\le \frac{1}{\chi _{m}}\Vert u_{m}(t)\Vert ^{2}_{2} \end{aligned}$$

(5.23)

where $\chi _{m}=\left( 1-C(N,\mu , r)\Lambda _{rq}^{-2q}\left( \frac{2q\theta p}{2q-\theta p}E(u_{0m})\right) ^{\frac{2q-\theta p}{\theta p}}\right)$. Using (5.9) and by a similar argument to that used in proving (4.17) we have

$$\begin{aligned} u_{m}\rightarrow u \;\; \text {in} \;\;\;C([0,T], L^{\nu }(\mathbb {R}^{N})), \;\;\forall \nu \in [2, p^{*}_{s}). \end{aligned}$$

(5.24)

On the other hand, we infer from $E(u_{0m})\rightarrow E(u_{0})$ as $m\rightarrow \infty$ that

$$\begin{aligned} \chi _{m}\rightarrow \left( 1-C(N,\mu , r)\Lambda _{rq}^{-2q}\left( \frac{2q\theta p}{2q-\theta p}E(u_{0})\right) ^{\frac{2q-\theta p}{\theta p}}\right) \;\; \text {as}\;\; m\rightarrow \infty . \end{aligned}$$

(5.25)

Combining (5.24) with (5.25) and passing to the limit in (5.23) as m goes to infinity, we conclude

$$\begin{aligned} \int _{t}^{T}\Vert u(t)\Vert _{2}^{\theta p}\,dt\le \frac{1}{\chi }\Vert u(t)\Vert ^{2}_{2}\;\; \;\;\forall t\ge 0, \end{aligned}$$

(5.26)

where $\chi =\left( 1-C(N,\mu , r)\Lambda _{qr}^{-2q}\left( \frac{2q\theta p}{2q-\theta p}E(u_{0m})\right) ^{\frac{2q-\theta p}{\theta p}}\right)$. Letting $T\rightarrow +\infty$ in (5.26) and using Lemma 5.1, we infer that

$$\begin{aligned} \Vert u(t)\Vert _{2}\le \left( \frac{\theta p}{1+\chi (\theta p-1)t}\right) ^{\frac{1}{\theta p-1}}, \;\;\forall t\ge 0. \end{aligned}$$

(5.27)

Now, we can apply (5.19) with $s_{1}=s$, $s_{2}=0$, $\eta =\frac{s-\epsilon }{s}$, $p_{2}=p_{1}=p$, to find

$$\begin{aligned} \Vert u(t)\Vert _{W^{s-\epsilon , p}(\mathbb {R}^{N})}\le C\Vert u(t)\Vert ^{\eta }_{W^{s,p}(\mathbb {R}^{N})}\Vert u(t)\Vert ^{1-\eta }_{p}\le \Vert u(t)\Vert ^{\eta }\Vert u(t)\Vert ^{1-\eta }_{p}. \end{aligned}$$

(5.28)

From (5.9) and (4.16), it follows that

$$\begin{aligned} \Vert u(t)\Vert \le \liminf _{m\rightarrow \infty } \Vert u_{m}(t)\Vert \le \left( \frac{2q\theta p}{2q-\theta p}\right) ^{\frac{1}{\theta p}}, \;\;\forall t\ge 0. \end{aligned}$$

(5.29)

Since $2< p< p^{*}_{s}$, then by using the classical interpolation inequality and the continuous embedding $W\hookrightarrow L^{p^{}_{s}}(\mathbb {R}^{N})$ we obtain

$$\begin{aligned} \Vert u(t)\Vert _{p}\le \Vert u(t)\Vert _{2}^{\beta }\Vert u(t)\Vert _{p^{*}_{s}}^{1-\beta }\le \Lambda _{p^{*}_{s}}^{\beta -1}\Vert u(t)\Vert _{2}^{\beta }\Vert u(t)\Vert ^{1-\beta }. \end{aligned}$$

(5.30)

where $\beta \in (0,1)$ satisfies that

$$\begin{aligned} \frac{1}{p}=\frac{\beta }{2}+\frac{1-\beta }{p^{*}_{s}}. \end{aligned}$$

By (5.27), (5.28), (5.29) and (5.30), we deduce

$$\begin{aligned}&\Vert u(t)\Vert _{W^{s-\epsilon , p}(\mathbb {R}^{N})}\le C\Vert u(t)\Vert ^{\eta }_{W^{s,p}(\mathbb {R}^{N})}\Vert u(t)\Vert ^{1-\eta }_{p}\\&\quad \le \Vert u(t)\Vert ^{\eta +(1-\eta ) (1-\beta )} \left( \frac{\theta p}{1+\chi (\theta p-1)t}\right) ^{\frac{\beta (1-\eta )}{\theta p-1}}\\&\quad \le \Lambda _{p^{*}_{s}}^{(\beta -1)(1-\eta )}\left( \frac{2q\theta p}{2q-\theta p}\right) ^{\frac{\eta +(1-\eta ) (1-\beta )}{\theta p}}\left( \frac{\theta p}{1+\chi (\theta p-1)t}\right) ^{\frac{\beta (1-\eta )}{\theta p-1}}, \end{aligned}$$

for all $t \ge 0$. This completes the proof. $\square$

6 Blow-up Phenomena

In this section we prove that the local weak solutions of initial value problem (1.1) blow up in finite time. Moreover, we give an estimate for the lower and upper bounds of the blow-up time.

In order to find an upper bound estimate for the blow-up time we need the following lemma, which is found in [44, Lemma 2.2].

Lemma 6.1

Suppose that a positive, twice differentiable function $\psi$ satisfies the inequality

$$\begin{aligned} \psi ''(t)\psi (t)-(1+\varsigma )(\psi ' (t))^{2}\ge 0 \end{aligned}$$

where $\varsigma >0$. If $\psi (0)> 0$, $\psi '(0)> 0$, then $\psi (t)\rightarrow +\infty$ as $t\rightarrow t_{*}$ and $t_{*}\le \frac{\psi (0)}{\varsigma \psi '(0)}$.

Proof

(Theorem 1.5) Hereafter, $T=T_{\max }$ is the maximal existence time of solutions to (1.1). Note that due to (1.20) we have $u_{0}\ne 0$ and

$$\begin{aligned} \Vert u_{0}\Vert > \left( \frac{2q\Lambda _{rq}^{2q}}{\theta pC(N,\mu ,r)}\right) ^{\frac{1}{2q-\theta p}}. \end{aligned}$$

(5.1)

Indeed, by the Hardy–Littlewood–Sobolev inequality

$$\begin{aligned} \frac{1}{\theta p}\Vert u_{0}\Vert ^{\theta p}< \frac{1}{2q}C(N,\mu ,r)\Vert u_{0}\Vert _{qr}^{2q}\le \frac{1}{2q\Lambda _{rq}^{2q}}C(N,\mu ,r)\Vert u_{0}\Vert ^{2q}, \end{aligned}$$

which implies (5.2) holds. Moreover, from (1.12) and the condition that $E(u_{0})<0$ we have $E(u(t))< 0$ for all $t\in [0, T)$. Thus in a similar way to (5.1) we infer that

$$\begin{aligned} \Vert u(t)\Vert > \left( \frac{2q\Lambda _{rq}^{2q}}{\theta pC(N,\mu ,r)}\right) ^{\frac{1}{2q-\theta p}}, \;\;\forall t\in [0,T). \end{aligned}$$

(5.2)

• Upper bound Define

  $$\begin{aligned} I(t)=\frac{1}{2}\int _{0}^{t}\Vert u(s)\Vert ^{2}_{2}\,ds+\frac{T-t}{2}\Vert u_{0}\Vert _{2}^{2}+\beta (\sigma +t)^{2}, \;\;\; 0<t< T. \end{aligned}$$

  where $\beta ,\sigma >0$ are to be determined. In view of the regularity of the weak local solutions stated in Definition 1.1 and [33, Proposition 1.2] we have

  $$\begin{aligned} I''(t)= \int _{\mathbb {R}^{N}}u_{t}(t)u(t)\,dx+2\beta . \end{aligned}$$

  Taking $u=v$ as a test function in Definition 1.1 yields

  $$\begin{aligned} I''(t)= -\Vert u(t)\Vert ^{\theta p}+\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}(t))u^{q}(t)\,dx+2\beta . \end{aligned}$$

  According to the definition of the energy functional

  $$\begin{aligned} E(u(t)) =\frac{1}{\theta p}\Vert u(t)\Vert ^{\theta p}-\frac{1}{2q}\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}(t))u^{q}(t)\,dx, \end{aligned}$$

  we have

  $$\begin{aligned} I''(t)= -\theta pE(u(t))+\left( 1-\frac{\theta p}{2q}\right) \int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}(t))u^{q}(t)\,dx+2\beta . \end{aligned}$$

  Since $2q> \theta p$, then the last equality implies

  $$\begin{aligned} I''(t)\ge -(\lambda +1) E(u(t))+2\beta , \end{aligned}$$

  where $\lambda =\theta p-1>0.$ In view of (1.12), we have

  $$\begin{aligned} I''(t)\ge (\lambda +1) \int _{0}^{t}\Vert u_{s}(t)\Vert ^{2}\,ds-(\lambda +1) E(u_{0})+2\beta . \end{aligned}$$

  (5.3)

  Since

  $$\begin{aligned} I'(t)= & {} \frac{1}{2}\int _{\mathbb {R}^{N}}u^{2}(s)\,dxds-\frac{1}{2}\int _{\mathbb {R}^{N}}u^{2}_{0}\,dx+2\beta (\sigma +t)\\= & {} \int _{0}^{t}\int _{\mathbb {R}^{N}}u_{s}(s)u(s)\,dxds+2\beta (\sigma +t). \end{aligned}$$

  It is follows that,

  $$\begin{aligned}{}[I'(t)]^{2}\le & {} 2 \int _{0}^{t}\Vert u_{s}(s)\Vert _{2}^{2}\,ds\int _{0}^{t}\Vert u(s)\Vert _{2}^{2}\,ds+8\beta ^{2} (\sigma +t)^{2}\nonumber \\\le & {} \left( 4\int _{0}^{t}\Vert u_{s}(s)\Vert _{2}^{2}\,ds+8\beta \right) I(t). \end{aligned}$$

  (5.4)

  Combining the above estimates, we find that for $\nu >0$

  $$\begin{aligned}&I''(t)I(t)-\frac{(\lambda +1)}{4}[I'(t)]^{2} \\&\quad \ge I(t)\left[ \Big ((\lambda +1)-(\lambda +1)\Big )\int _{0}^{t}\Vert u_{s}(s)\Vert _{2}^{2}\,ds-(\lambda +1) E(u_{0})-2\beta \lambda \right] \\&\quad = I(t)\left[ -(\lambda +1) E(u_{0})-2\beta \lambda \right] \ge 0, \end{aligned}$$

  provided $\beta \in \Big (0, \frac{-\theta p E(u_{0})}{2(\theta p-1)}\Big ]$. Since $\theta \in (2, N/(N-sp))$ and $p>2$ it follows that $\theta p>4$. Invoking Lemma 6.1, yields

  $$\begin{aligned} \lim _{t\rightarrow T^{-}}I(t)=+\infty , \end{aligned}$$

  (5.5)

  and

  $$\begin{aligned} T\le \frac{4I(0)}{(\theta p-4)I'(0)}=\frac{2(T\Vert u_{0}\Vert _{2}^{2}+2\beta \sigma ^{2})}{2\beta \sigma (\theta p-4)}=\frac{\Vert u_{0}\Vert _{2}^{2}}{\beta \sigma (\theta p-4)}T+\frac{\sigma }{(\theta p-4)}. \end{aligned}$$

  This inequality can be rewritten as

  $$\begin{aligned} T\left( 1- \frac{\Vert u_{0}\Vert _{2}^{2}}{\beta \sigma (\theta p-4)}\right) \le \frac{\sigma }{(\theta p-4)}. \end{aligned}$$

  (5.6)

  for any $\beta \in \Big (0, \frac{-\theta p E(u_{0})}{2(\theta p-1)}\Big ]$ and $\sigma > 0$. Now if we choose $\sigma \in \left( \frac{\Vert u_{0}\Vert _{2}^{2}}{\beta (\theta p-4)}, +\infty \right)$ we infer that

  $$\begin{aligned} 0<\frac{\Vert u_{0}\Vert _{2}^{2}}{\sigma \beta (\theta p-4)}< 1. \end{aligned}$$

  This combined with (5.6) yields

  $$\begin{aligned} T\le \frac{\sigma }{(\theta p-4)} \left( 1- \frac{\Vert u_{0}\Vert _{2}^{2}}{\beta \sigma (\theta p-4)}\right) ^{-1}=\frac{\beta \sigma ^{2}}{\beta \sigma (\theta p-4)-\Vert u_{0}\Vert _{2}^{2}}. \end{aligned}$$

  (5.7)

  In what follows for the sake of simplicity we put $\kappa =\frac{-\theta p E(u_{0})}{2(\theta p-1)}$. Let us consider the following set

  $$\begin{aligned} \Sigma= & {} \left\{ (\beta ,\sigma ):\,\beta \in (0, \kappa ],\; \sigma \in \left( \frac{\Vert u_{0}\Vert _{2}^{2}}{\beta (\theta p-4)}, +\infty \right) \right\} \\= & {} \left\{ (\beta ,\sigma ):\, \sigma \in \left( \frac{\Vert u_{0}\Vert _{2}^{2}}{\kappa (\theta p-4)}, +\infty \right) ,\; \beta \in \Big (\frac{\Vert u_{0}\Vert _{2}^{2}}{\sigma (\theta p-4)}, \kappa \Big ] \right\} \end{aligned}$$

  In view of (5.7), we have

  $$\begin{aligned} T\le \inf _{(\beta , \sigma )\in \Sigma } \frac{\beta \sigma ^{2}}{\beta \sigma (\theta p-4)-\Vert u_{0}\Vert _{2}^{2}}. \end{aligned}$$

  Set

  $$\begin{aligned} g(\sigma , \tau )=\frac{\tau \sigma }{\tau (\theta p-4)-\Vert u_{0}\Vert _{2}^{2}}. \end{aligned}$$

  where $\tau =\beta \sigma$. Through straightforward computation one can show that $g(\sigma ,\tau )$ is decreasing in $\tau$. Thus, it follows that

  $$\begin{aligned} T\le \inf _{\sigma \in \left( \frac{\Vert u_{0}\Vert _{2}^{2}}{\kappa (\theta p-4)}, +\infty \right) } \frac{\kappa \sigma ^{2}}{ \sigma \kappa (\theta p-4)-\Vert u_{0}\Vert _{2}^{2}}. \end{aligned}$$

  (5.8)

  Minimizing the right hand side of (5.8) one has

  $$\begin{aligned} T\le \frac{4\Vert u_{0}\Vert ^{2}_{2}}{\kappa (\theta p-4)^{2}}. \end{aligned}$$

  (5.9)

  Now by substituting the value of $\kappa$ in (5.9) we get

  $$\begin{aligned} T\le \frac{8\Vert u_{0}\Vert ^{2}_{2}(\theta p-1)}{-\theta pE(u_{0}) (\theta p-4)^{2}}. \end{aligned}$$

  (5.10)

  According to the definition of I and (5.5), we have

  $$\begin{aligned} \lim _{t\rightarrow T^{-}}\Vert u(t)\Vert ^{2}_{2}=+\infty . \end{aligned}$$

  (5.11)

• Lower bound Now we turn to prove the second part. Taking $u=v$ as a test function in Definition 1.1 we get

  $$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^{N}}u^{2}(t)\,dx=-2\Vert u(t)\Vert ^{\theta p}+2\int _{\mathbb {R}^{N}}(\mathcal {K}_{\mu }*u^{q}(t))u^{q}(t)\,dx. \end{aligned}$$

  Using the Hardy–Littlewood–Sobolev inequality we deduce

  $$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^{N}}u^{2}(t)\,dx\le -2\Vert u(t)\Vert ^{\theta p}+2C(N,\mu ,r)\Vert u(t)\Vert ^{2q}_{rq}. \end{aligned}$$

  Since $2< rq<p^{*}_{s}$, it follows from the interpolation inequality that

  $$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^{N}}u^{2}(t)\,dx\le -2\Vert u(t)\Vert ^{\theta p}+2C(N,\mu ,r)\Vert u(t)\Vert ^{2q\gamma }_{2}\Vert u(t)\Vert ^{2q(1-\gamma )}_{p^{*}_{s}}, \end{aligned}$$

  where $\gamma \in (0,1)$ satisfies

  $$\begin{aligned} \frac{1}{qr}=\frac{\gamma }{2}+\frac{1-\gamma }{p^{*}_{s}}. \end{aligned}$$

  Now, by using a similar argument that in the proof of Theorem 1.1, for any $\varepsilon >0$ we have

  $$\begin{aligned} \frac{d}{dt}\int _{\mathbb {R}^{N}}u^{2}(t)\,dx\le -2\Vert u(t)\Vert ^{\theta p}+C(\varepsilon )\left( \Vert u(t)\Vert _{2}^{2}\right) ^{\alpha }+\varepsilon \Vert u(t)\Vert ^{\theta p} \end{aligned}$$

  (5.12)

  where

  $$\begin{aligned} \alpha =\frac{\theta pq\gamma }{\theta p-2q(1-\gamma )}>1, \end{aligned}$$

  and

  $$\begin{aligned} C(\varepsilon )=\varepsilon ^{-\frac{2\theta ^{2} p^{2}}{\theta p-2q(1-\gamma )}}\Lambda ^{-\frac{2q(1-\gamma )\theta p}{\theta p-2q(1-\gamma )}}_{p^{*}_{s}}. \end{aligned}$$

  Put

  $$\begin{aligned} g(t)=\int _{\mathbb {R}^{N}}u^{2}(t)\,dx=\Vert u(t)\Vert ^{2}_{2}. \end{aligned}$$

  Then (5.12) can be rewritten as

  $$\begin{aligned} \frac{d g(t)}{dt}\le -2\Vert u(t)\Vert ^{\theta p}+C(\varepsilon )g^{\alpha }(t)+\varepsilon \Vert u(t)\Vert ^{\theta p}. \end{aligned}$$

  Taking $\varepsilon =2$ we arrive at inequality

  $$\begin{aligned} \frac{d g(t)}{dt}\le Cg^{\alpha }(t). \end{aligned}$$

  Integrating this inequality from 0 to t yields

  $$\begin{aligned} \frac{1}{C}\int _{g(0)}^{g(t)}\frac{1}{\tau ^{\alpha }}\,d\tau \le t. \end{aligned}$$

  Thus, it turns out that

  $$\begin{aligned} \frac{1}{C(1-\alpha )}g^{1-\alpha }(t)+\frac{1}{C(\alpha -1)}g^{1-\alpha }(0)\le t, \end{aligned}$$

  which together with $\lim \limits _{t\rightarrow T^{-}} g(t)=\lim \limits _{t\rightarrow T^{-}}\Vert u(t)\Vert ^{2}_{2}=+\infty$ and $\alpha >1$ give

  $$\begin{aligned} \frac{1}{C(\alpha -1)}g^{1-\alpha }(0) \le T, \end{aligned}$$

  (5.13)

  where $g(0)=\Vert u_{0}\Vert ^{2}_{2}> 0$ and $C=2^{-\frac{2\theta ^{2} p^{2}}{\theta p-2q(1-\gamma )}}\Lambda ^{-\frac{2q(1-\gamma )\theta p}{\theta p-2q(1-\gamma )}}_{p^{*}_{s}}$. Hence from (5.10) and (5.13) we reach the conclusion.

$\square$