A Liouville-type theorem for an integral system on a half-space $R_{+}^n$

By a Liouville-type theorem, we here mean the statement of nonexistence of nontrivial (bounded or not) solutions on the whole space or on a half-space. In the last two decades, Liouville-type theorems have been widely used, in conjunction with rescaling arguments, to derive a priori estimates for solutions of boundary value problems.

For $r>\frac{n}{n-\alpha }$ ($0<\alpha <n$), we obtained the following Liouville-type theorem.

Theorem 1.1 [1]

Suppose $r>\frac{n}{n-\alpha }$. If the solution u of (1.1) satisfies $u\in L^{\frac{n(r-1)}{\alpha }}(R_{+}^n)$ and is nonnegative, then $u\equiv 0$.

This is the so-called critical case.

In [2] Zhuo and Li discussed regularity and rotational symmetry of solutions for integral system (1.2).

Theorem 1.2 [2]

Let $(u,v)$ be a pair of positive solutions of (1.2) with $p,q\ge 1$. Assume that $u\in L^{p+1}(R_{+}^n)$ and $v\in L^{q+1}(R_{+}^n)$, then every positive solution $(u,v)$ of (1.2) is rotationally symmetric about some line parallel to $x_n$-axis.

where α is an even number.

Theorem 1.3 [2]

Let $(u,v)$ be a pair of solutions of (1.2) up to a constant, then $(u,v)$ satisfies (1.3).

In this paper, we use a simple and more general method to derive that the solution pair $(u,v)$ of (1.2) is strictly monotonically increasing with respect to the variable $x_n$ and further present the nonexistence of positive solutions of (1.2) under some integrability conditions.

Theorem 1.4 Let $(u,v)$ be a pair of positive solutions of (1.2) with $p,q\ge 1$. Assume that $u\in L^{p+1}(R_{+}^n)$ and $v\in L^{q+1}(R_{+}^n)$, then both u and v are strictly monotonically increasing with respect to the variable $x_n$.

Theorem 1.5 Let $(u,v)$ be a pair of positive solutions of (1.2) with $p,q\ge 1$. Assume that $u\in L^{p+1}(R_{+}^n)$ and $v\in L^{q+1}(R_{+}^n)$ are nonnegative, then $u=v\equiv 0$.

where $x^{\ast }=(x_1,\dots ,x_{n-1},-x_n)$ is a reflection of the point x about the $\partial R_{+}^n$.

the complement of ${\mathrm{\Sigma }}_{\lambda }$ in $R_{+}^n$.

be a reflection of the point $x=(x_1,x_2,\dots ,x_n)$ about the plane $T_{\lambda }$.

The following lemma states some properties of the function $G(x,y)$. Here we present a proof.

In fact,

Then, for $s,t>0$, we have

(2.7)

(2.8)

(2.9)

1. (i)

   From (2.4), (2.5), (2.7) and (2.8), we obtain (2.1).

    

Then (2.3) follows immediately from (2.7) and (2.8).

This completes the proof of Lemma 2.1. □

Remark The properties of the function $G(x,y)$ defined on a half-space are very similar to the properties of Green’s function for a poly-harmonic operator on the ball with Dirichlet boundary conditions. One could find this interesting relation from [3, 4] and [5].

In this section, by using the method of moving planes in integral forms, we derive the nonexistence of positive solutions to integral system (1.2) and obtain a new Liouville-type theorem on a half-space. To prove the theorems, we need several lemmas.

Lemma 3.1 Let $(u,v)$ be any pair of positive solutions of (1.2). For any $x\in {\mathrm{\Sigma }}_{\lambda }$, we have

(3.1)

(3.2)

Proof

Obviously, we have

Similarly, we could derive the second inequality in the lemma. This completes the proof of Lemma 3.1. □

Proof of Theorem 1.4 To prove Theorem 1.4, we compare $(u(x),v(x))$ and $(u_{\lambda }(x),v_{\lambda }(x))$ on ${\mathrm{\Sigma }}_{\lambda }$. The proof consists of two steps.

In the second step, we will move our plane $T_{\lambda }$ toward the positive direction of $x_n$-axis as long as inequality (3.3) holds.

Now, inequality (3.7) implies ${\parallel u_{\lambda }-u\parallel }_{L^{p+1}({\mathrm{\Sigma }}_{\lambda }^u)}=0$, and therefore ${\mathrm{\Sigma }}_{\lambda }^u$ must be measure zero. Similarly, one can show that ${\mathrm{\Sigma }}_{\lambda }^v$ is measure zero. Therefore, (3.3) holds. This completes Step 1.

Step 2. (Move the plane to the limiting position to derive symmetry and monotonicity.)

Now, by (3.9), we have ${\parallel u_{\lambda }-u\parallel }_{L^{p+1}({\mathrm{\Sigma }}_{\lambda }^u)}=0$, therefore ${\mathrm{\Sigma }}_{\lambda }^u$ must be measure zero. Similarly, ${\mathrm{\Sigma }}_{\lambda }^v$ must also be measure zero. This verifies (3.8), therefore both $u(x)$ and $v(x)$ are symmetric about the plane $T_{{\lambda }_0}$. Also, the monotonicity easily follows from the argument. This completes the proof of Theorem 1.4. □

Proof of Theorem 1.5 To prove the theorem, firstly we will show that the plane cannot stop at $x_n={\lambda }_0$ for some ${\lambda }_0<+\mathrm{\infty }$, that is, we will prove that ${\lambda }_0=+\mathrm{\infty }$.

Suppose that ${\lambda }_0<+\mathrm{\infty }$, the process of Theorem 1.4 shows that the plane $x_n=2{\lambda }_0$ is the symmetric points of the boundary $\partial R_{+}^n$ with respect to the plane $T_{{\lambda }_0}$, and we derive that $u(x)=0$ and $v(x)=0$ when x is on the plane $x_n=2{\lambda }_0$. This contradicts the pair of positive solutions $(u(x),v(x))$ of (1.2), thus ${\lambda }_0=+\mathrm{\infty }$.

Besides, we know that both $u(x)$ and $v(x)$ of positive solutions of (1.2) are strictly monotonically increasing in the positive direction of $x_n$-axis, but $u\in L^{p+1}(R_{+}^n)$ and $v\in L^{q+1}(R_{+}^n)$, so we come to the conclusion that the pair of positive solutions $(u(x),v(x))$ of (1.2) does not exist.

This completes the proof of Theorem 1.5. □