1. Introduction
In this paper, we study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for the following system involving fractional Laplacian
where $s_1,s_2\in (0,1)$, $n>\max \{2s_1,2s_2\}$, $\mathbb {R}_+^n=\{x=(x',x_n)\in \mathbb {R}^n|x_n>0\}$ and $x'=(x_1,x_2,\ldots,x_{n-1})$.
The fractional Laplacian $(-\Delta )^s \, (0< s<1)$ is a nonlocal operator defined by
where $P.V.$ stands for the Cauchy principal value and $C(n,s)=(\int _{\mathbb {R}^n}\frac {1-\cos \xi }{|\xi |^{n+2s}}\,{\rm d}\xi )^{-1}$ (see [Reference Caffarelli and Silvestre2, Reference Di Nezza, Palatucci and Valdinoci11]). Let
Then for $u\in L_{2s}, (-\Delta )^su$ can be defined in distributional sense (see [Reference Silvestre34])
Moreover, $(-\Delta )^su$ is well defined for $u\in L_{2s}\cap C_{loc}^{1,1}(\mathbb {R}^n)$. We call $(u,v)$ a classical solution of (1.1) if $(u,v)\in ( L_{2s_1}\cap C^{1,1}_{loc}(\mathbb {R}_+^n)\cap C(\mathbb {R}^n)) \times (L_{2s_2}\cap C^{1,1}_{loc}(\mathbb {R}_+^n)\cap C(\mathbb {R}^n))$ and satisfies (1.1).
As is well known, the method of moving planes and moving spheres play an important role in proving the nonexistence of solutions. Chen et al. [Reference Chen, Li and Li3, Reference Chen, Li and Zhang4] introduced a direct method of moving planes and moving spheres for fractional Laplacian, which have been widely applied to derive the symmetry, monotonicity and nonexistence and even a prior estimates of solutions for some equations involving fractional Laplacian. In such process, some suitable forms of maximum principles are the key ingredients. The method of moving planes in integral forms is also a vital tool for classification of solutions (see [Reference Chen, Li and Ou5]).
Recently, Li and Zhuo [Reference Li and Zhuo20] classified anti-symmetric classical solutions of Lane–Emden system (1.1) in the case of $s_1=s_2=:s\in (0,1)$ and $\alpha =\beta =0$. They established the following Liouville type theorem.
Proposition 1.1 ([Reference Li and Zhuo20])
Given $0< p,q\leq \frac {n+2s}{n-2s},$ assume that $(u, v)$ is an anti-symmetric classical solution of system (1.1). If $0 < pq < 1$ or $p + 2s > 1$ and $q + 2s > 1$, then $(u,v)\equiv (0,0)$.
As a corollary of proposition 1, the nonexistence results in the larger space $L_{2s+1}$ follows immediately for the case $p+2s>1$, $q+2s>1$.
Proposition 1.2 ([Reference Li and Zhuo20])
Assume that $u$ and $v \in L_{2s+1}\cap C^{1,1}_{loc}(\mathbb {R}^n_+)\cap C(\mathbb {R}^n)$ satisfy system (1.1). Then if $0 < p, q \leq \frac {n+2s}{n-2s}$, $p+2s > 1$ and $q + 2s > 1$, $(u,v)\equiv (0,0)$ is the only solution.
The nonexistence of anti-symmetric classical solutions to the corresponding scalar problem were given in [Reference Zhuo and Li37].
The following Hardy–Hénon system with homogeneous Dirichlet boundary conditions has been investigated widely
There are enormous nonexistence results of (1.2) for the case $\Omega =\mathbb {R}^n$. We list some main results as follows.
If $s_1=s_2=1$, for $\alpha,\beta \geq 0$, system (1.2) is the well-known Hénon–Lane–Emden system. It has been conjectured that the Sobolev's hyperbola
is the critical dividing curve between existence and nonexistence of solutions to (1.2). Particularly, the Hénon–Lane–Emden conjecture states that system (1.2) admits no nonnegative non-trivial solutions if $p>0$, $q>0$ and $\frac {n+\alpha }{p+1}+\frac {n+\beta }{q+1}>n-2$. For $\alpha =\beta =0$, this conjecture has been completely proved for radial solutions (see [Reference Mitidieri23, Reference Serrin and Zou32]). However, for non-radial solutions, the conjecture is only fully answered when $n\leq 4$ (see [Reference Poláčik, Quittner and Souplet29, Reference Serrin and Zou33, Reference Souplet35]). In higher dimensions, the conjecture was partially solved. Figueiredo and Felmer [Reference de Figueiredo and Felmer14] showed that system (1.2) admits no classical positive solutions if
Busca and Manásevich [Reference Busca and Manásevich1] proved the conjecture if
where
When $\alpha,\beta >0$, Fazly and Ghoussoub [Reference Fazly and Ghoussoub13] showed that the conjecture holds for dimension $n = 3$ under the assumption of the boundedness of positive solutions, Li and Zhang [Reference Li and Zhang22] removed this assumption and proved this conjecture for dimension $n = 3$. When $\min \{\alpha,\beta \}>-2$, the conjecture is proved for bounded solutions in $n=3$ (see [Reference Phan27]).
If $s_1=s_2=:s\in (0,1), \alpha, \beta \geq 0$, there are fewer nonexistence results of solutions to system (1.2) in the case of $p>0$, $q>0$ and $\frac {n+\alpha }{p+1}+\frac {n+\beta }{q+1}>n-2s$, namely the case that $(p,q)$ locates at bottom left of the fractional Sobolev's hyperbola $\left \{p>0,q>0:\frac {n+\alpha }{p+1}+\frac {n+\beta }{q+1}=n-2s\right \}$. For $\alpha =\beta =0$, Quaas and Xia in [Reference Quaas and Xia30] proved that there exist no classical positive solutions to (1.2) provided that
where
Note that region (1.3) of $(p,q)$ contains the following region:
As $\min \{\alpha,\beta \}>-2s$, Peng [Reference Peng26] derived that system (1.2) admits no nonnegative classical solutions if $0< p<\frac {n+2s+2\alpha }{n-2s}$ and $0< q<\frac {n+2s+2\beta }{n-2s}$.
For scalar equation (i.e. $s_1=s_2:=s, \alpha =\beta, p=q, u=v$), in the Laplacian case, if $\alpha =0$, a celebrated Liouville type theorem was showed by Gidas and Spruck [Reference Gidas and Spruck17] for $1< p<\frac {n+2}{n-2}$; if $\alpha \leq -2$ and $p>1$, there is no any positive solution (see [Reference Gidas and Spruck16, Reference Ni25]); if $\alpha >-2$ and $1< p<\frac {n+2+2\alpha }{n-2}$, Phan and Souplet [Reference Phan and Souplet28] derived a Liouville theorem for bounded solutions; if $0 < p \leq 1$, the nonexistence result was proved by Dai and Qin [Reference Dai and Qin8] for any $\alpha$. We also refer to [Reference Gabriele15, Reference Mitidieri and Pokhozhaev24] and references therein. For the scalar fractional Laplacian case, Chen et al. [Reference Chen, Li and Li3], Jin et al. [Reference Jin, Li and Xiong18] proved the nonexistence results for $\alpha =0$ and $0< p<\frac {n+ 2s}{n-2s}.$ If $\alpha >-2s$, Dai and Qin [Reference Dai and Qin9] showed a Liouville type theorem for optimal range $0< p<\frac {n+2s+2\alpha }{n-2s}$.
For the case $\Omega =\mathbb {R}^n_+$, the following several main results exist.
If $s_1=s_2=1$, $\alpha =\beta =0$, $\min \{p,q\}>1$, a Liouville theorem is proved for bounded solutions by Chen et al. [Reference Chen, Lin and Zou6]. If $s_1=s_2=:s \in (0,1)$, for $\alpha =\beta =0$, the nonexistence of positive viscosity-bounded solutions to system (1.2) was showed by Quaas and Xia in [Reference Quaas and Xia31]. For $\alpha,\beta >-2s$, if $p\geq \frac {n+2s+\alpha }{n-2s}$ and $q\geq \frac {n+2s+\beta }{n-2s}$, Duong and Le [Reference Duong and Le12] obtained the nonexistence of solutions satisfying the following decay at infinity:
For general $s_1, s_2\in (0,1)$, Le in [Reference Le19] concluded a Liouville type theorem. Precisely, they obtained that if $1\leq p\leq \frac {n+2s_1+2\alpha }{n-2s_2}$, $1\leq q\leq \frac {n+2s_2+2\alpha }{n-2s_1}$ and $(p,q)\neq (\frac {n+2s_1+2\alpha }{n-2s_2},\frac {n+2s_2+2\alpha }{n-2s_1}$), $\alpha >-2s_1$ and $\beta >-2s_2$, then $(u,v)\equiv (0,0)$ is the only nonnegative classical solutions to system (1.2). More nonexistence results for general nonlinearities in a half space can be seen in [Reference Dai and Peng10, Reference Zhang, Yu and He36]. For the corresponding scalar problem of (1.2) with Laplacian and $\alpha =\beta =0$, Gidas and Spruck [Reference Gidas and Spruck17] obtained the nonexistence of nontrivial nonnegative classical solution of (1.2) for $1< p\leq \frac {n+2}{n-2}$. For the corresponding scalar problem (1.2) with fractional Laplacian and $\alpha =\beta =0$, Chen et al. [Reference Chen, Li and Li3] showed that $u\equiv 0$ is the only nonnegative solution to (1.2) for $1< p\leq \frac {n+2s}{n-2s}$. Recently, a Liouville type theorem of the corresponding scalar problem for $1< p<\frac {n+2s+2\alpha }{n-2s}, \alpha >-2s$ and $s\in (0,1]$ was established by Dai and Qin in [Reference Dai and Qin9].
In this paper, we will study nonexistence of anti-symmetric classical solutions to system (1.1) for general $\alpha, \beta, p, q$.
For $\alpha >-2s_1$ and $\beta >-2s_2$, we denote
Note that for the case $s_1=s_2$, the set $\mathcal {R}_{sub}$ locates at bottom left of the preceding fractional Sobolev's hyperbola.
Throughout this paper, we always assume $s_1,s_2\in (0,1)$, $n>\max \{2s_1,2s_2\}$ and use $C$ to denote a general positive constant whose value may vary from line to line even the same line. Our main results are as follows.
Theorem 1.1 For $(p,q)\in \mathcal {R}_{sub}$, assume that $(u,v)$ is a classical solution of system (1.1). For either one of the following two cases:
(i) $\min \{p+2s_1,p+2s_1+\alpha \}>1$ and $\min \{q+2s_2,q+2s_2+\beta \}>1$,
(ii) $0< pq<1$ with $\alpha \geq -2s_1pq$, $\beta \geq -2s_2pq$,
we have that $(u,v)=(0,0)$.
The nonexistence results (i) of theorem 1 can be extended to a larger space.
Theorem 1.2 Assume that $(p,q)\in \mathcal {R}_{sub}$ and $(u,v)\in (L_{2s_1+1}\cap C^{1,1}_{loc}(\mathbb {R}^n_+)\cap C(\mathbb {R}^n))\times (L_{2s_2+1}\cap C^{1,1}_{loc}(\mathbb {R}^n_+)\cap C(\mathbb {R}^n))$ satisfies system (1.1). Then for the case that $\min \{p+2s_1,p+2s_1+\alpha \}>1$ and $\min \{q+2s_2,q+2s_2+\beta \}>1$, $(u,v)\equiv (0,0)$ is the only solution.
Combining our anti-symmetric property, in the proof of theorem 1.1, we only utilized the extended spaces $L_{2s_1+1}, L_{2s_2+1}$ instead of the usual spaces $L_{2s_1}, L_{2s_2}$ in the case that $\min \{p+2s_1,p+2s_1+\alpha \}>1$ and $\min \{q+2s_2,q+2s_2+\beta \}>1$. One can see that theorem 1.2 is a direct corollary of (i) of theorem 1.1.
Remark 1.1 Our results of theorems 1.1 and 1.2 are the extension to general $s_1,s_2,\alpha,\beta$ of the nonexistence results of Li and Zhuo [Reference Li and Zhuo20] (see preceding propositions 1.1 and 1.2) except one critical point of $(p,q)$.
Remark 1.2 When $s_1=s_2$, $p=q$, $\alpha =\beta$ and $u=v$, the results of theorems 1.1 and 1.2 are the nonexistence of nontrivial classical solutions to the corresponding scalar problem.
Under appropriate decay conditions of $u$ and $v$ at infinity, we can extend the nonexistence result of classical solutions of (1.1) to an unbounded domain of $(p,q)$. Particularly, this unbounded domain, except at most a bounded sub-domain, locates at above the preceding fractional Sobolev's hyperbola for the case $s_1=s_2$.
Theorem 1.3 Suppose $p\geq \frac {n+2s_1+\alpha }{n-2s_2}$, $q\geq \frac {n+2s_2+\beta }{n-2s_1}$, $\alpha >-2s_2$, $\beta >-2s_1$. Assume $(u,v)$ is a classical solution of system (1.1) satisfying
for some $C>0$, where $a=-\frac {2s_1+2s_2+\beta }{q-1}$ and $b=-\frac {2s_1+2s_2+\alpha }{p-1}$. Then $(u,v)\equiv (0,0)$.
Remark 1.3 The results of theorem 1.3 are new even if for the corresponding scalar problem with $\alpha =0$.
Remark 1.4 When $\alpha,\beta$ are positive, define the region
Note that $\mathcal {R}_{sup}$ is contained in the nonexistence region of $(p,q)$ obtained in theorem 1.1. Hence, if $(p,q)\in \mathcal {R}_{sup}$, theorem 1.1 tells us that the results of theorem 1.3 still hold true without the decay conditions.
2. Preliminaries
In this section, we introduce and prove some necessary lemmas.
Proposition 2.1 ([Reference Duong and Le12])
Let $s\in (0,1)$ and $w(y)\in L_{2s}\cap C_{loc}^{1,1}(\mathbb {R}^n)$ satisfy $w(y)=-w(y^-_\lambda )$, where $y^-_\lambda =(y', 2\lambda -y_n)$ for any real number $\lambda$. Assume there exists $x\in \Sigma _\lambda$ such that
where $\Sigma _\lambda =\{x\in \mathbb {R}^n|x_n<\lambda \}$. Then we have the following claims:
(i) if
\[\mathop{\lim\inf}\limits_{|x|\to \infty }|x|^{2s}c(x)\geq0, \]there exists a constant $R_0>0$ (depending on c, but independent of $w$) such that\[ |x|< R_0, \](ii) if $c$ is bounded below in $\Sigma _\lambda$, there exists a constant $\ell >0$ (depending on the lower bound of $c$, but independent of $w$) such that
\[ {\rm d}(x,T_\lambda)> \ell, \]where $T_\lambda =\{x\in \mathbb {R}^n|x_n=\lambda \}$.
We want to point out that the constant $\ell$ is non-increasing about $\lambda$, since $\ell$ is non-decreasing about the lower bound of $c$, which can be seen from the proof of proposition 2.1 in [Reference Duong and Le12].
In order to apply the method of moving planes to prove the nonexistence, we need to establish the following estimate.
Lemma 2.1 Let $s\in (0,1)$ and $w(y)\in L_{2s}\cap C_{loc}^{1,1}(\mathbb {R}^n)$ satisfy $w(y)=-w(y^-_\lambda )$. Assume there exists $x\in \Sigma _\lambda$ such that $w(x)=\mathop {\inf }\nolimits _{\Sigma _\lambda }w(y)<0$. Then we have
where $d=d(x,T_\lambda )$ and the constant $C(n,s)$ is positive.
Proof. Applying the definition of fractional Laplacian, we have
By an elementary calculation (see [Reference Cheng, Huang and Li7]), we derive
Hence, combining this and (2.1), we complete the proof of lemma 2.1.
In order to apply the method of moving spheres to prove the nonexistence, we need to establish a similar estimate as that of lemma 2.1. To this end, we need to introduce some notations. For any real number $\lambda >0$, we denote
Let $x^\lambda =\frac {\lambda ^2x}{|x|^2}$ be the inversion of the point $x=(x', x_n)$ about the sphere $S_\lambda$ and $x^*=(x', -x_n)$. Denote
Lemma 2.2 Let $w(x)\in L_{2s}\cap C_{loc}^{1,1}(\mathbb {R}_+^n)$ satisfy
Assume there exists $\tilde {x}\in B_\lambda ^+$ such that $w(\tilde {x})=\inf _{B_\lambda ^+}w(x)<0$. Then we have
where $h_\lambda (\tilde {x},y)=\frac {1}{|\tilde {x}-y|^{n+2s}}-\frac {1}{|\frac {|y|}{\lambda }\tilde {x}-\frac {\lambda }{|y|}y|^{n+2s}}+ \frac {1}{|\frac {|y|}{\lambda }\tilde {x}-\frac {\lambda }{|y|}y^*|^{n+2s}}-\frac {1}{|\tilde {x}-y^*|^{n+2s}}>0$ for $\tilde {x},y\in B_\lambda ^+$, $\delta =\min \{\tilde {x}_n,\lambda -|\tilde {x}|\}$ and $C(n,s)$ is a positive constant.
Proof. By the definition of fractional Laplacian and assumptions (2.2), we derive
Using similar arguments in [Reference Li and Zhuo21], we can obtain that $h_\lambda (x,y)>0$ for $x,y\in B_\lambda ^+$.
Furthermore, choose $r<\tilde {x}_n$ small such that $H:=\{x\in B_\delta (\tilde {x})|x_n>\tilde {x}_n\}\subset \{x\in \mathbb {R}^n|x_n>\tilde {x}_n\}\subset (B_r^+(0))^C$ where $\delta =\min \{\tilde {x}_n,\lambda -|\tilde {x}|\}$, then we calculate
From the definition $\delta =\min \{\tilde {x}_n,\lambda -|\tilde {x}|\}$, we have $|\tilde {x}-y^*|< C\tilde {x}_n$ for any $y\in B_\delta ^+(\tilde {x})$. Simple calculations imply that
It is easy to see that
Lemma 2.3 Let $\alpha,\beta >-n$. Suppose that $(u,v)$ is a nonnegative classical solution for the following system:
Then for $x\in \mathbb {R}^n_+$, we have
where $C$ is a positive constant.
Proof. Define a cut-off function $\eta (x)\in C_0^{\infty }(\mathbb {R}^n)$ satisfying $\eta (x)=0$ for $|x|>1$ and $\eta (x)=1$ for $|x|<\frac {1}{2}$. Denote $\eta _R(x)=\eta (\frac {x}{R})$ for large $R$ and
Note that $(u_R(x), v_R(x))$ is a solution for the following system:
Let $U_R(x)=u(x)-u_R(x)$ and $V_R(x)=v(x)-v_R(x)$. From (2.7) and (2.8), we derive
By the definitions of $U_R(x)$ and $V_R(x)$, obviously, for $x\in \mathbb {R}^n_+$,
where we used the assumptions $\alpha,\beta >-n$.
Next, we claim that $U_R(x)\geq 0$ and $V_R(x)\geq 0$ for $x\in \mathbb {R}_+^n$. If not, from (2.10) we know that there exists some $\hat {x}\in \mathbb {R}_+^n$ such that $U_R(\hat {x})=\mathop {\inf }\nolimits _{\mathbb {R}_+^n}U_R(x)<0$. Then,
This leads a contradiction with the first equation in (2.9). Thus, $U_R(x)\geq 0$ holds true for any $x\in \mathbb {R}_+^n$, that is, $u(x)\geq u_R(x)$ in $\mathbb {R}_+^n$. Letting $R \to \infty$, we obtain
Similarly, one has
3. Proof of theorem 1.1
In this section, we are ready to prove theorem 1.1. For (i) of theorem 1.1, namely the case $\min \{p+2s_1,p+2s_1+\alpha \}>1$ and $\{q+2s_2,q+2s_2+\beta \}>1$, we use the method of moving spheres to derive a lower bound for $u(x)$ and $v(x)$. Then, lemma 2.3 and a ‘bootstrap’ iteration process will give the better lower-bound estimates which can imply the nonexistence result. For (ii) of theorem 1.1, namely the case $0< pq<1$, $\alpha \geq -2s_1pq$ and $\beta \geq -2s_2pq$, a direct application of lemma 2.3 and iteration technique may give its proof.
3.1 Proof of (i) of theorem 1.1
Proof. By contradiction, assume that $(u,v) \not \equiv (0,0)$, then we can derive that $u>0$ and $v>0$ in $\mathbb {R}^n_+$. Indeed, if there exists some $\hat {x}\in \mathbb {R}_+^n$ such that $u(\hat {x})=0$, from the anti-symmetry of $u$, we have
which contradicts with the equation
Thus $u(x)>0,$ and using the same arguments as above, we easily obtain $v(x)>0$. Therefore, we may assume that $u(x)>0$ and $v(x)>0$ in the rest proof of (i) of theorem 1.
Let $u_\lambda (x)$ and $v_\lambda (x)$ be the Kelvin transform of $u(x)$ and $v(x)$ centred at origin, respectively
for arbitrary $x\in \mathbb {R}^n\setminus \{0\}$. By an elementary calculation, $u_\lambda (x)$ and $v_\lambda (x)$ satisfy the following system:
where
Note that both $\tau _1$ and $\tau _2$ are nonnegative and they will not be zero simultaneously, since $(p,q)\in \mathcal {R}_{sub}$.
Denote
By elementary calculations and the mean value theorem, for $x\in B_\lambda ^+$, there holds
where $\xi _\lambda (x)$ is between $v(x)$ and $v_\lambda (x)$, $\eta _\lambda (x)$ is between $u(x)$ and $u_\lambda (x)$. Note that
Next, we will use the method of moving spheres to claim that $U_\lambda (x)\geq 0$ and $V_\lambda (x)\geq 0$ in $B_\lambda ^+$ for any $\lambda >0$.
Step 1. Give a start point. We show that for sufficiently small $\lambda >0$,
Suppose (3.4) is not true, there must exist a point $\bar {x}\in B_\lambda ^+$ such that at least one of $U_\lambda (\bar {x})$ and $V_\lambda (\bar {x})$ is negative at this point. Without loss of generality, we assume
We will obtain contradictions for all four possible cases, respectively.
Case 1. $(p,q)\in \mathcal {R}_{sub}$ and $p\geq 1, q\geq 1$. Due to $p\geq 1$, by the convexity of the function $f(t)=t^p$, then we can take $\xi _\lambda (x)=v(x)$ in (3.1). From equation (3.1) and lemma 2.2, we have
Hence,
If $\delta =\min \{\lambda -|\bar {x}|,\bar {x}_n\}=\lambda -|\bar {x}|$, which implies that $\lambda -|\bar {x}|\leq \bar {x}_n\leq |\bar {x}|$, using the fact and (3.6), we obtain
As $\lambda \to 0$, the right-hand side of (3.7) will go to infinity since $\alpha >-2s_1$. This is impossible.
If $\delta =\min \{\lambda -|\bar {x}|,\bar {x}_n\}=\bar {x}_n$, from $\bar {x}_n\leq |\bar {x}|$ and (3.6), we derive
which is also impossible.
Case 2. $0< p,q<1$. Due to $p<1$, we can take $\xi _\lambda (x) =v_\lambda (x)$. From equation (3.1) and lemma 2.2, we have
Analogous to (3.7) and (3.8), there holds
or
Applying lemma 2.3 and the mean value theorem, we obtain that for $x\in B_1^+$,
For $x\in (B_1^+)^C$, we derive
Similarly, we have
Then by the definition of $v_\lambda (x)$, we obtain
If $\delta =\lambda -|\bar {x}|\leq \bar {x}_n$, that is, $|\bar {x}|\geq \frac {\lambda }{2}$, combining (3.11) and (3.12), we conclude that for $\bar {x}\in B_\lambda ^+$ and sufficiently small $\lambda$,
which gives that
Due to $\min \{p+2s_1, p+2s_1+\alpha \}>1$ and $\frac {\lambda }{|\bar {x}|}\leq 2$, inequality (3.13) is impossible as $\lambda >0$ sufficiently small.
If $\delta =\bar {x}_n$, it follows from (3.10) and (3.12) that
which implies that
Either one of the two inequalities will yield a contradiction since the left terms go to infinity as $\min \{p+2s_1,p+2s_1+\alpha \}>1$ and $\lambda >0$ small enough.
For case 3: $(p,q)\in \mathcal {R}_{sub}, p\geq 1, 0< q<1$ and case 4: $(p,q)\in \mathcal {R}_{sub}, 0< p<1, q\geq 1$, similar argument as that of cases 1 and 2 can show that $U_\lambda (x)\geq 0$ and $V_\lambda (x)\geq 0$ in $B_\lambda ^+$ for sufficiently small $\lambda >0$. Therefore, (3.4) holds.
Step 2. Now we move the sphere $S_\lambda$ outwards as long as (3.4) holds. Define
We will show that $\lambda _0=+\infty$. Suppose on the contrary that $0<\lambda _0<+\infty$. We want to show that there exists some small $\varepsilon >0$ such that for any $\lambda \in (\lambda _0,\lambda _0+\varepsilon )$,
This implies that the plane $S_{\lambda _0}$ will be moved outwards a little bit further, which contradicts with the definition of $\lambda _0$.
Firstly, we claim that
Indeed, if there exists some point $x^0\in B_{\lambda _0}^+$ such that $U_{\lambda _0}(x^0)= 0$, we have
On the other hand, it is easy to get that
If $\tau _1>0$, then $(-\Delta )^{s_1}U_{\lambda _0}(x^0)>0$, where we use the facts that $V_{\lambda _0}\geq 0$ and $v>0$ in $\mathbb {R}^n_+$. If $\tau _1=0$, then we have that $\tau _2>0$. Moreover, $V_{\lambda _0}(x^0)=0$ follows from (3.16) and (3.17). Using an argument similar to (3.16) and (3.17), we derive
which is absurd. Thus, $U_{\lambda _0}(x)> 0$ is proved. Similarly, we derive that $V_{\lambda _0}(x)> 0$. Hence, (3.15) holds.
Next, we will show that the sphere can be moved further outwards. The continuity of $u(x)$ and (3.15) yield that there exists some sufficiently small $l\in (0,\frac {\lambda _0}{2})$ and $\varepsilon _1\in (0,\frac {\lambda _0}{2})$ such that for $\lambda \in (\lambda _0,\lambda _0+\varepsilon _1)$,
For $x\in B_\lambda ^+\setminus B_{\lambda _0-l}^+$, using the similar proof of (3.4), we can deduce that
Note that the distance between $\bar {x}$ and $S_\lambda$, i.e. $\lambda -|\bar {x}|$, plays an important role in this process.
Hence, it follows from (3.18) and (3.19) that for all $\lambda \in (\lambda _0,\lambda _0+\varepsilon _1)$,
Similarly, we can also prove that there exists $\varepsilon _2>0$ such that for all $\lambda \in (\lambda _0, \lambda _0+\varepsilon _2)$,
Let $\varepsilon =\min \{\varepsilon _1,\varepsilon _2\}$, therefore, (3.14) can be completely concluded. This contradicts with the definition of $\lambda _0$. So $\lambda _0=+\infty$.
Then, we have for every $\lambda >0$,
which gives that,
For any given $|x|\geq 1$, let $\lambda =\sqrt {|x|}$, then it follows from (3.20) that
and similarly from (3.21), we obtain
Now we make full use of the above properties to derive some lower-bound estimates of solutions to (1.1) through iteration technique.
Let $\theta _0=\frac {n-2s_1}{2}+1$, $\sigma _0=\frac {n-2s_2}{2}+1$. From lemma 2.3, inequality (3.23) and the mean value theorem, we have for $x_n>1$,
Similarly, we have
Denote $\theta _1=p(\sigma _0-1)-(\alpha +2 s_1)+1$ and $\sigma _1=q(\theta _0-1)-(\beta +2s_2)+1$. Repeat the above process replacing (3.23) by (3.25), then we have
and analogously,
where $\theta _2=p(\sigma _1-1)-\alpha -2s_1+1$ and $\sigma _2=q(\theta _1-1)-\beta -2s_2+1$.
After such $k$ iteration steps, we derive
where $\theta _{k+1}=p(\sigma _{k}-1)-\alpha -2s_1+1$ and $\sigma _{k+1}=q(\theta _{k}-1)-\beta -2s_2+1$. Elementary calculations give that
where $m=0,1,2,\ldots$.
For the case $pq\geq 1$, we claim that both $\{\theta _k\}$ and $\{\sigma _k\}$ are decreasing sequences and unbounded from below. Denote
Note that $A_e,A_o\leq 0$ and they will not be zero simultaneously, since $(p,q)\in \mathcal {R}_{sub}$. By elementary calculations, we have
Hence, the sequence $\{\theta _k\}$ is decreasing.
Combining the above properties of $A_e, A_o$, the assumption $pq\geq 1$ and (3.29), we deduce that $\theta _k\to -\infty$ as $k\to +\infty$. Similarly, we can show that $\{\sigma _k\}$ is also decreasing and $\sigma _k\to -\infty$. These and (3.27) indicate that $u(x)$ and $v(x)$ do not belong to any $L_{2s}$. Hence, the solutions $(u,v)\equiv (0,0)$ when $pq\geq 1$.
Next, we consider the case $pq<1$. From (3.28), we can conclude that
Since $p+2s_1+\alpha >1$ and $q+2s_2+\beta >1$, we have
Hence, from (3.30) and (3.27), we have for $x_n>1$,
Combining this with lemma 2.3, we have
Let $x'-y'=(x_n+y_n)z'$, we have
Let $y_n=x_nz_n$, one has
Due to $p+2s_1+\alpha >1$ and $q+2s_2+\beta >1$, we have $\frac {q(2s_1+\alpha )+(2s_2+\beta )}{1-pq}>1$, which implies that
This yields that the right-hand side of (3.32) is infinity, which is impossible. Hence, for $pq<1$ we also obtain $(u,v)\equiv (0,0).$
In sum, we conclude that there is no nontrivial classical solutions to system (1.1) for the case that $(p,q)\in \mathcal {R}_{sub}$, $\min \{p+2s_1, p+2s_1+\alpha \}>1$, $\min \{q+2s_2,q+2s_2+\beta \}>1$.
3.2 Proof of (ii) of theorem 1.1
Proof. Assume that $(u,v) \not \equiv (0,0)$, from the proof of (i), we know that $u>0$ and $v>0$ in $\mathbb {R}_+^n$. Applying lemma 2.3, for $x_n>\min \{2,\frac {|x'|}{10}\}$, we have
Similarly, for $x_n>\min \{2,\frac {|x'|}{10}\}$, we have
Iterating (3.33) with (3.34), for $x_n>\min \{2,\frac {|x'|}{10}\}$, we get
Using the same argument as that of (3.35), for $x_n>\min \left \{2,\frac {|x'|}{10}\right \}$, we obtain
After $k$ iteration steps, it is easy to see that for $|x|$ large and $x_n>\min \left \{2,\frac {|x'|}{10}\right \}$,
Here,
where
Simple calculations imply that
where $m=0,1,2,\ldots.$
From $0< pq<1$, we obtain
This yields that for $|x|$ large,
as $k\to +\infty$. Due to $0< pq<1$ and $\alpha \geq -2s_1pq$, $\beta \geq -2s_2pq$, we have
This contradicts with the assumptions that $u(x)\in L_{2s_1}$ and $v(x)\in L_{2s_2}$. Hence, $(u,v)\equiv (0,0)$.
4. Proof of theorem 1.3
In this section, we give the proof of theorem 1.3 by using the method of moving planes.
Proof. Suppose on the contrary that $(u,v) \not \equiv (0,0)$, we know that $u>0$ and $v>0$ in $\mathbb {R}_+^n$. Let $\bar {u}(x)$ and $\bar {v}(x)$ be the Kelvin transform of $u(x)$ and $v(x)$ centred at origin, respectively
for arbitrary $x\in \mathbb {R}^n\setminus \{0\}$. Then, $\bar {u}(x)$ and $\bar {v}(x)$ satisfy the following system:
where
Note that $\bar {\tau }_1\leq 0$ and $\bar {\tau }_2\leq 0$ due to $p\geq \frac {n+2s_1+\alpha }{n-2s_2}$ and $q\geq \frac {n+2s_2+\beta }{n-2s_1}$. Obviously, for $|x|$ large enough,
For any real number $\rho >0$ and $x\in \Sigma _\rho$, define
where $\Sigma _\rho$, $T_\rho$ and $x^-_\rho$ are defined as in proposition 2.1. Then, for $x\in \Sigma _\rho \cap \mathbb {R}^n_+$, we have
where we use the fact $p>1$ in the last inequality. Similarly,
Step 1. We claim that for $\rho >0$ sufficiently small,
Otherwise, from (4.1), there exists some $\bar {x}\in \Sigma _\rho \cap \mathbb {R}_+^n$ such that at least one of $\bar {U}_\rho (x)$, $\bar {V}_\rho (x)$ is negative. Without loss of generality, we may assume that
Combining equation (4.2) and lemma 2.1, we deduce
This yields that
Observe that (4.1) and the decay conditions of $u$ and $v$ in theorem 1.3 ensure that
where we used the assumption $\alpha >-2s_2$. Hence, inequality (4.6) is impossible as $\rho >0$ is sufficiently small. Therefore, (4.4) holds.
Step 2. Move the plane $T_\rho$ upwards along the $x_n$-axis as long as (4.4) holds. Let
We will show that $\rho _0=+\infty$ by contradiction arguments.
Suppose on the contrary that $0<\rho _0<+\infty$. We will verify that
Then using the above equalities (4.7), we immediately obtain
which is impossible. Thus $\rho _0=+\infty$ must hold.
Therefore, our goal is to prove (4.7). Suppose that (4.7) does not hold, then we deduce that
Otherwise, there exists some point $\tilde {x}\in \Sigma _{\rho _0}\cap \mathbb {R}^n_+$ such that $\bar {U}_{\rho _0}(\tilde {x})=0$. We have
On the contrary, it is easy to get that
where we use the fact $\bar {V}_{\rho _0}\geq 0$. This leads to a contradiction. Hence, (4.8) holds.
Now we show that the plane $T_{\rho _0}$ can be moved upwards a little bit further and hence obtain a contradiction with the definition of $\rho _0$. Precisely, we will verify that there exists some small $\varepsilon >0$ such that for any $\rho \in (\rho _0,\rho _0+\varepsilon )$,
where $\varepsilon$ is determined later.
If (4.9) is not true, then for any $\varepsilon _k\to 0$ as $k\to +\infty$, there exists $\rho _k\in (\rho _0,\rho _0+\varepsilon _k)$ and $x_k\in \mathbb {R}_+^n\cap \Sigma _{\rho _k}$ such that
Similar argument as that of (4.5) gives that
where $c(x)=-|x|^{-\bar {\tau }_1} p\bar {v}^{p-1}(x)$. From (4.1) and the decay conditions of $u$ and $v$, we deduce that
where we used the assumption $\alpha >-2s_2$. Then from proposition 2.1 we know that there exists $\ell _k>0$ and $R_0>0$ such that
Denote $\ell _0 (>0)$ as the constant given in proposition 2.1 corresponding to the half space $\Sigma _{\rho _0+1}$. Combining the remark about the monotonicity of $\ell$ with respect to $\lambda$ below proposition 2.1, (4.13) and the fact that $\varepsilon _k\to 0$, we have that
If $\rho _0-\frac {\ell _0}{2}\leq 0$, then (4.14) contradicts with the fact that $x_k\in \mathbb {R}^n_+$. If $\rho _0-\frac {\ell _0}{2}>0$, due to (4.8) and continuity of $\bar {u}$, we know that there exists $\varepsilon '\in (0,\frac {\ell _0}{2})$ such that for any $\varepsilon _k\leq \varepsilon '$ and $\rho \in (\rho _0,\rho _0+\varepsilon _k)$,
This contradicts with (4.14) and (4.10). Hence, we derive that for any $\rho \in (\rho _0, \rho _0+\varepsilon ')$ with $\varepsilon '>0$ small enough,
Similarly, we may verify that there exists $\varepsilon ''>0$ such that for any $\rho \in (\rho _0, \rho _0+\varepsilon '')$ the inequality holds
Let $\varepsilon =\min \{\varepsilon ',\varepsilon ''\}$, then (4.9) follows immediately and hence (4.7) holds, which yields that $\rho _0=+\infty$.
The result $\rho _0=+\infty$ indicates that both $\bar {u}(x)$ and $\bar {v}(x)$ are monotone increasing along the $x_n$-axis. This contradicts with the asymptotic behaviours (4.1). Therefore, $(\bar {u},\bar {v})=(0,0)$, which yields that $(u,v)=(0,0)$. We complete the proof of theorem 1.3.
Acknowledgements
The authors are supported by the Natural Science Foundation of Hunan Province, China (Grant No. 2022JJ30118).