Theory and Modern Applications

# Nonexistence of positive solutions of an integral system with weights

## Abstract

In this article, we study nonexistence, radial symmetry, and monotonicity of the positive solutions for a class of integral systems with weights. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle. Our results are new even for the Laplace case.

2010 MSC: 35J99; 45E10; 45G05.

## 1. Introduction

In this article, we study positive solutions of the following system of integral equations in N (N ≥ 3),

$\left\{\begin{array}{c}\hfill u\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{v{\left(y\right)}^{q}}{\mid y{\mid }^{\xi }\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \\ \hfill v\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{u{\left(y\right)}^{p}}{\mid y{\mid }^{\eta }\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \end{array}\right\$
(1.1)

with ξ, η < 0, 0 < α < N, $1 and $1. Under certain restrictions of regularity, the non-negative solution (u, v) of (1.1) is proved to be trivial or radially symmetric with respect to some point of N respectively.

The integral system (1.1) is closely related to the system of PDEs in N

$\left\{\begin{array}{c}{\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}u=\frac{{v}^{q}}{\mid x{\mid }^{\xi }},\\ {\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}v=\frac{{u}^{p}}{\mid x{\mid }^{\eta }}.\end{array}\right\$
(1.2)

In fact, every positive smooth solution of PDE (1.2) multiplied by a constant satisfies (1.1). This equivalence between integral and PDE systems for α = 2 can be verified as in the proof of Theorem 1 in . For single equations, we refer to [, Theorem 4.1]. Here, in (1.2), the following definition is used.

${\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}u={\left(\mid \chi {\mid }^{\alpha }{u}^{\wedge }\right)}^{\vee }$

where is the Fourier transformation and its inverse.

When α = 2, Figueiredo et al.  studied the system of PDEs (1.2) in a bounded smooth domain Ω with Dirichlet boundary conditions. They found a critical hyperbola, given by

$\frac{N-\xi }{q+1}+\frac{N-\eta }{p+1}=N-2,\phantom{\rule{1em}{0ex}}p,\phantom{\rule{2.77695pt}{0ex}}q>0.$
(1.3)

Below this hyperbola they showed the existence of nontrivial solutions of (1.2). Interestingly, this hyperbola is closely related to the problem (1.2) in the whole space. For α = 2 and ξ, η = 0, i.e., the elliptic systems without weights in N , Serrin conjectured that (1.2) has no bounded positive solutions below the hyperbola of (1.3). It is known that above this hyperbola, (1.2) has positive solutions. Some Liouville-type results were shown in [4, 5] (see also [6, 7]).

When α = 2 and ξ, η ≤ 0, Felmer  proved the radial symmetry of the solutions of the corresponding elliptic system (1.2) by the moving plane method which was based on Maximum Principle, going back to Alexandroff, Serrin , and Gidas et al. .

For ξ, η > 0, Chen and Li  proved the radial symmetry of solutions of (1.1) on the hyperbola (1.3). In the special case, when ξ, η = 0, the system (1.1) reduces to

$\left\{\begin{array}{c}\hfill u\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{v{\left(y\right)}^{q}}{\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \\ \hfill v\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{u{\left(y\right)}^{p}}{\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y.\hfill \end{array}\right\$
(1.4)

The integral system (1.4) is closely related to the system of PDEs

$\left\{\begin{array}{c}{\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}u={v}^{q},\\ {\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}v={u}^{p}.\end{array}\right\$
(1.5)

Recently, using the method of moving planes, Ma and Chen  proved a Liouville-type theorem of (1.4), and for the more generalized system,

$\left\{\begin{array}{c}\hfill u\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{v{\left(y\right)}^{q}}{\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \\ \hfill v\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{u{\left(y\right)}^{p}}{\mid x-y{\mid }^{N-\beta }}\mathsf{\text{d}}y.\hfill \end{array}\right\$
(1.6)

Huang et al.  proved the existence, radial symmetry and monotonicity under some assumptions of p, q, α, and β. Furthermore, using Doubling Lemma indicated in , which is an extension of an idea of Hu , Chen and Li [, Theorem 4.3] obtained the nonexistence of positive solutions of (1.4) under some stronger integrability conditions (e.g., $u,v\in {L}_{loc}^{\mathrm{\infty }}$ are necessary). In fact, for System (1.5) of α = 2, Liouville-type theorems are known for (q, p) in the region $\mathsf{\text{[0,}}\frac{N+2}{N-2}\right]\phantom{\rule{2.77695pt}{0ex}}×\left[\mathsf{\text{0,}}\frac{N+2}{N-2}\right]$. For the interested readers, we refer to [17, 18] and their generalized cases [19, 20], where the results were proved by the moving plane method or the method of moving spheres which both deeply depend on Maximum Principle. In , Mitidieri proved that if (q, p) satisfies

$\frac{1}{p+1}+\frac{1}{q+1}>\frac{N-2}{N},\phantom{\rule{1em}{0ex}}p,\phantom{\rule{2.77695pt}{0ex}}q>0,$
(1.7)

then System (1.5) possesses no nontrivial radial positive solutions. Later, Mitidieri  showed that a Liouvillle-type theorem holds if (q, p) satisfies

$\frac{N-2}{N}\le max\left\{\frac{q+1}{qp-1},\frac{p+1}{qp-1}\right\},$

generalizing a work by Souto . In , Serrin and Zou proved that for (q, p) satisfying (1.7), there exists no positive solution of System (1.1) when the solution has an appropriate decay at infinity.

When α = 2, it has been conjectured that a Liouville-type theorem of System (1.5) holds if the condition (1.7) holds. This conjecture is further suggested by the works of Van der Vorst  and Mitidieri  on existence in bounded domains, Hulshof and Van der Vorst , Figueiredo and Felmer  on existence on bounded domains through variational method, and Serrin and Zou  on existence of positive radial solutions when the inequality in (1.7) is reversed. Figueierdo and Felmer , Souto , and Serrin and Zou  studied System (1.5) and obtained some Liouville-type results. Ma and Chen  gave a partial generalized result about their work. Serrin conjectured that if (q, p) satisfies (1.7), System (1.5) has no bounded positive solutions. It is known that outside the region of (1.7), System (1.5) has positive solutions. We believe that the critical hyperbola in the conjecture is closely related to the famous Hardy-Littlewood-Sobolev inequality  and its generalization. For more results about elliptic systems, one may look at the survey paper of Figueierdo .

There are some related works about this article. When u(x) = v(x) and $q=p=\frac{N+\alpha }{N-\alpha }$, System (1.4) becomes the single equation

(1.8)

The corresponding PDE is the well-known family of semilinear equations

(1.9)

In particular, when N ≥ 3 and α = 2, (1.9) becomes

(1.10)

The classification of the solutions of (1.10) has provided an important ingredient in the study of the well-known Yamabe problem and the prescribing scalar curvature problem. Equation (1.10) was studied by Gidas et al. , Caffarelli et al. , Chen and Li  and Li . They classified all the positive solutions. In the critical case, Equation (1.10) has a two-parameter family of solutions given by

$u\left(x\right)={\left(\frac{c}{d\phantom{\rule{2.77695pt}{0ex}}+\mid x-\stackrel{̄}{x}{\mid }^{2}}\right)}^{\frac{N-2}{2}},$
(1.11)

where $c={\left[N\left(N-2\right)d\right]}^{\frac{1}{2}}$ with d > 0 and $\stackrel{̄}{x}\in {ℝ}^{N}$. Recently, Wei and Xu  generalized this result to the solutions of the more general Equation (1.9) with α being any even number between 0 and N.

Apparently, for other real values of α between 0 and N, (1.9) is also of practical interest and importance. For instance, it arises as the Euler-Lagrange equation of the functional

$I\left(u\right)=\underset{{ℝ}^{N}}{\int }\mid {\left(-\mathrm{\Delta }\right)}^{\frac{\alpha }{4}}u{\mid }^{2}\mathsf{\text{d}}x∕{\left(\underset{{ℝ}^{N}}{\int }\mid u{\mid }^{\frac{2N}{N-\alpha }}\mathsf{\text{d}}x\right)}^{\frac{N-\alpha }{N}}.$

The classification of the solutions would provide the best constant in the inequality of the critical Sobolev imbedding from ${H}^{\frac{\alpha }{2}}\left({ℝ}^{N}\right)$ to ${L}^{\frac{2N}{N-\alpha }}\left({ℝ}^{N}\right)$:

${\left(\underset{{ℝ}^{N}}{\int }\mid u{\mid }^{\frac{2N}{N-\alpha }}\mathsf{\text{d}}x\right)}^{\frac{N-\alpha }{N}}\le C\underset{{ℝ}^{N}}{\int }\mid {\left(-\mathrm{\Delta }\right)}^{\frac{\alpha }{4}}u{\mid }^{2}\mathsf{\text{d}}x.$

Let us emphasize that considerable attention has been drawn to Liouville-type results and existence of positive solutions for general nonlinear elliptic equations and systems, and that numerous related works are devoted to some of its variants, such as more general quasilinear operators and domains, and the blowup questions for nonlinear parabolic equations and systems. We refer the interested reader to [20, 22, 26, 27, 3639], and some of the references therein.

Our results in the present article can be considered as a generalization of those in [8, 12, 17, 18]. We note that we here use the Kelvin-type transform and a new type of moving plane method introduced by Chen-Li-Ou, and our new ingredient is the use of Hardy-Littlewood-Sobolev inequality instead of Maximum Principle. Our results are new even for the Laplace case of α = 2.

Our main results are the following two theorems.

Theorem 1.1. Let the pair (u, v) be a non-negative solution of (1.1) and $\frac{N-\eta }{N-\alpha }, $\frac{N-\xi }{N-\alpha } with ξ, η < 0 and 0 < α < N, but $p=\frac{N+\alpha -\eta }{N-\alpha }$ and $q=\frac{N+\alpha -\xi }{N-\alpha }$ are not true at the same time. Moreover, assume that $u\in {L}_{loc}^{\beta }\left({ℝ}^{N}\right)$ and $v\in {L}_{loc}^{\varphi }\left({ℝ}^{N}\right)$ with $\beta =\frac{p-1}{\frac{\left(N-\alpha \right)p+\eta }{N}-1}$ and $\varphi =\frac{q-1}{\frac{\left(N-\alpha \right)q+\xi }{N}-1}$. Then both u and v are trivial, i.e., (u, v) = (0, 0).

Theorem 1.2. Let the pair (u, v) be a non-negative solution of (1.1) and $p=\frac{N+\alpha -\eta }{N-\alpha }$, $q=\frac{N+\alpha -\xi }{N-\alpha }$ with ξ, η < 0 and 0 < α < N. Moreover, assume that $u\in {L}_{loc}^{\beta }\left({ℝ}^{N}\right)$ and $v\in {L}_{loc}^{\varphi }\left({ℝ}^{N}\right)$ with $\beta =\frac{\left(2\alpha -\eta \right)N}{\alpha \left(N-\alpha \right)}$ and $\varphi =\frac{\left(2\alpha -\xi \right)N}{\alpha \left(N-\alpha \right)}$. Then, u and v are radially symmetric and decreasing with respect to some point of N .

Remark 1.1. Due to the technical difficulty, we here only consider the nonexistence and symmetry of positive solutions in the range of ξ, η < 0, $p>\frac{N-\eta }{N-\alpha }$ and $q>\frac{N-\xi }{N-\alpha }$. For ξ, η > 0, Chen and Liproved the radial symmetry of solutions of (1.1) on the hyperbola (1.3). For ξ = η = 0 and max{1, 2/(N - 2)} < p, q < ∞, Chen and Li [, Theorem 4.3] obtained the nonexistence of positive solutions of (1.1) under some stronger integrability conditions (e.g., $u,v\in {L}_{loc}^{\mathrm{\infty }}$ are necessary). We note that there exist many open questions on nonexistence and symmetry of positive solutions of the equation with weights as (1.1) in the rest range of p, q, ξ, and η. It is an interesting research subject in the future.

We shall prove Theorem 1.1 via the Kelvin-type transform and the moving plane method (see [2, 40, 41]) and prove Theorem 1.2 by the similar idea as in .

Throughout the article, C will denote different positive constants which depend only on N, p, q, α and the solutions u and v in varying places.

## 2. Kelvin-type transform and nonexistence

In this section, we use the moving plane method to prove Theorem 1.1. First, we introduce the Kelvin-type transform of u and v as follows, for any x ≠ 0,

$ū\left(x\right)=\mid x{\mid }^{\alpha -N}u\left(\frac{x}{\mid x{\mid }^{2}}\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\stackrel{̄}{v}\left(x\right)=\mid x{\mid }^{\alpha -N}v\left(\frac{x}{\mid x{\mid }^{2}}\right).$

Then by elementary calculations, one can see that (1.1) and (1.2) are transformed into the following forms:

$\left\{\begin{array}{c}\hfill ū\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{\stackrel{̄}{v}{\left(y\right)}^{q}}{\mid y{\mid }^{s}\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \\ \hfill \stackrel{̄}{v}\left(x\right)=\underset{{ℝ}^{N}}{\int }\frac{ū{\left(y\right)}^{p}}{\mid y{\mid }^{t}\mid x-y{\mid }^{N-\alpha }}\mathsf{\text{d}}y,\hfill \end{array}\right\$
(2.1)

and

$\left\{\begin{array}{c}{\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}ū=\phantom{\rule{2.77695pt}{0ex}}\mid x{\mid }^{-s}{\stackrel{̄}{v}}^{q},\\ {\left(-\mathrm{\Delta }\right)}^{\alpha ∕2}\stackrel{̄}{v}=\phantom{\rule{2.77695pt}{0ex}}\mid x{\mid }^{-t}{ū}^{p},\end{array}\right\$
(2.2)

where t = (N + α) - η - (N - α)p ≥ 0 and s = (N + α) - ξ - (N - α)q ≥ 0. Obviously, both $ū\left(x\right)$ and $\stackrel{̄}{v}\left(x\right)$ may have singularities at origin. Since $u\in {L}_{loc}^{\beta }\left({ℝ}^{N}\right)$ and $v\in {L}_{loc}^{\varphi }\left({ℝ}^{N}\right)$, it is easy to see that $ū\left(x\right)$ and $\stackrel{̄}{v}\left(x\right)$ have no singularities at infinity, i.e., for any domain Ω that is a positive distance away from the origin,

$\underset{\mathrm{\Omega }}{\int }ū{\left(y\right)}^{\beta }\mathsf{\text{d}}y<\mathrm{\infty }\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\underset{\mathrm{\Omega }}{\int }\stackrel{̄}{v}{\left(y\right)}^{\varphi }\mathsf{\text{d}}y<\mathrm{\infty }.$
(2.3)

In fact, for y = z/|z|2, we have

$\begin{array}{cc}\hfill {\int }_{\mathrm{\Omega }}ū{\left(y\right)}^{\beta }\mathsf{\text{d}}y& =\underset{\mathrm{\Omega }}{\int }{\left(\mid y{\mid }^{\alpha -N}u\left(\frac{y}{\mid y{\mid }^{2}}\right)\right)}^{\beta }\mathsf{\text{d}}y\hfill \\ =\underset{{\mathrm{\Omega }}^{*}}{\int }{\left(\mid z{\mid }^{N-\alpha }u\left(z\right)\right)}^{\beta }\mid z{\mid }^{-2N}\mathsf{\text{d}}z\hfill \\ =\underset{{\mathrm{\Omega }}^{*}}{\int }\mid z{\mid }^{\beta \left(N-\alpha \right)-2N}u{\left(z\right)}^{\beta }\mathsf{\text{d}}z\hfill \\ \le C\underset{{\mathrm{\Omega }}^{*}}{\int }u{\left(z\right)}^{\beta }\mathsf{\text{d}}z\hfill \\ <\mathrm{\infty }.\hfill \end{array}$

For the second equality, we have made the transform y = z/|z|2. Since Ω is a positive distance away from the origin, Ω*, the image of Ω under this transform, is bounded. Also, note that β(N - α) - 2N > 0 by the assumptions of Theorem 1.1. Then, we get the estimate (2.3).

For a given real number λ, define

${\mathrm{\Sigma }}_{\lambda }=\left\{x=\left({x}_{1},\dots ,{x}_{n}\right)\mid {x}_{1}\ge \lambda \right\}.$

Let x λ = (2λ - x 1, x 2,..., x n ), ${ū}_{\lambda }\left(x\right)=ū\left({x}^{\lambda }\right)$ and ${\stackrel{̄}{v}}_{\lambda }\left(x\right)=\stackrel{̄}{v}\left({x}^{\lambda }\right)$.

The following lemma is elementary and is similar to Lemma 2.1 in .

Lemma 2.1. For any solution ($ū\left(x\right)$, $\stackrel{̄}{v}\left(x\right)$) of (2.1), we have

${ū}_{\lambda }\left(x\right)-ū\left(x\right)=\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\left(\mid x-y{\mid }^{\alpha -N}-\mid {x}^{\lambda }-y{\mid }^{\alpha -N}\right)\left[\mid {y}^{\lambda }{\mid }^{-s}{\stackrel{̄}{v}}_{\lambda }{\left(y\right)}^{q}-\mid y{\mid }^{-s}\stackrel{̄}{v}{\left(y\right)}^{q}\right]\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{d}}y$
(2.4)

and

${\stackrel{̄}{v}}_{\lambda }\left(x\right)-\stackrel{̄}{v}\left(x\right)=\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\left(\mid x-y{\mid }^{\alpha -N}-\mid {x}^{\lambda }-y{\mid }^{\alpha -N}\right)\left[\mid {y}^{\lambda }{\mid }^{-t}{ū}_{\lambda }{\left(y\right)}^{p}-\mid y{\mid }^{-t}ū{\left(y\right)}^{p}\right]\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{d}}y.$
(2.5)

Proof. It is easy to see that

$\begin{array}{cc}\hfill ū\left(x\right)& =\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\mid y{\mid }^{-s}\mid x-y{\mid }^{\alpha -N}{\stackrel{̄}{v}}^{q}\left(y\right)\mathsf{\text{d}}y\hfill \\ \phantom{\rule{1em}{0ex}}+\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\mid {y}^{\lambda }{\mid }^{-s}\mid {x}^{\lambda }-y{\mid }^{\alpha -N}{\overline{{v}_{\lambda }}}^{q}\left(y\right)\mathsf{\text{d}}y.\hfill \end{array}$
(2.6)

Substituting x by x λ , we have

$\begin{array}{cc}\hfill ū\left({x}^{\lambda }\right)& =\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\mid y{\mid }^{-s}\mid {x}^{\lambda }-y{\mid }^{\alpha -N}{\stackrel{̄}{v}}^{q}\left(y\right)\mathsf{\text{d}}y\hfill \\ \phantom{\rule{1em}{0ex}}+\underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\mid {y}^{\lambda }{\mid }^{-s}\mid x-y{\mid }^{\alpha -N}{\overline{{v}_{\lambda }}}^{q}\left(y\right)\mathsf{\text{d}}y.\hfill \end{array}$
(2.7)

The fact that |x - y λ | = |x λ - y| implies (2.4). Similarly, one can show that (2.5) holds. So, Lemma 2.1 is proved.

Proof of Theorem 1.1.

Outline: Let x 1 and x 2 be any two points in N . We shall show that

$u\left({x}_{1}\right)=u\left({x}_{2}\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}v\left({x}_{1}\right)=v\left({x}_{2}\right)$

and therefore u and v must be constants. This is impossible unless u = v = 0. To obtain this, we show that u and v are symmetric about the midpoint (x 1 + x 2)/2. We may assume that the midpoint is at the origin. Let $ū$ and $\stackrel{̄}{v}$ be the Kelvin-type transformations of u and v, respectively. Then, what left to prove is that $ū$ and $\stackrel{̄}{v}$ are symmetric about the origin. We shall carry this out in the following three steps.

Step 1. Define

${\mathrm{\Sigma }}_{\lambda }^{ū}=\left\{x\in {\mathrm{\Sigma }}_{\lambda }\mid ū\left(x\right)<{ū}_{\lambda }\left(x\right)\right\}$

and

${\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}=\left\{x\in {\mathrm{\Sigma }}_{\lambda }\mid \stackrel{̄}{v}\left(x\right)<{\stackrel{̄}{v}}_{\lambda }\left(x\right)\right\}$

We show that for sufficiently negative values of λ, both ${\mathrm{\Sigma }}_{\lambda }^{ū}$ and ${\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}$ must be empty.

Whenever x, y Σ λ , we have that |x - y| ≤ |x λ - y|. Moreover, since λ < 0, |y λ | |y| for any y Σ λ . Then by emma 2.1, for any x Σ λ , it is easy to verify that

$\begin{array}{cc}\hfill {ū}_{\lambda }\left(x\right)-ū\left(x\right)& \le \underset{{\mathrm{\Sigma }}_{\lambda }}{\int }\left(\mid x-y{\mid }^{\alpha -N}-\mid {x}^{\lambda }-y{\mid }^{\alpha -N}\right)\mid y{\mid }^{-s}\left[{\stackrel{̄}{v}}_{\lambda }{\left(y\right)}^{q}-\stackrel{̄}{v}{\left(y\right)}^{q}\right]\mathsf{\text{d}}y\hfill \\ \le \underset{{\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}}{\int }\mid x-y{\mid }^{\alpha -N}\mid y{\mid }^{-s}\left[{\stackrel{̄}{v}}_{\lambda }{\left(y\right)}^{q}-\stackrel{̄}{v}{\left(y\right)}^{q}\right]\mathsf{\text{d}}y\hfill \\ \le \underset{{\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}}{\int }\mid x-y{\mid }^{\alpha -N}\mid y{\mid }^{-s}\left[{\stackrel{̄}{v}}_{\lambda }{\left(y\right)}^{q-1}\left({\stackrel{̄}{v}}_{\lambda }\left(y\right)-\stackrel{̄}{v}\left(y\right)\right)\right]\mathsf{\text{d}}y.\hfill \end{array}$
(2.8)

Now we recall the double weighted Hardy-Littlewood-Sobolev inequality which was generalized by Stein and Weiss :

$\parallel \int \frac{f\left(y\right)}{\mid x{\mid }^{\gamma }\mid x-y{\mid }^{\lambda }\mid y{\mid }^{\tau }}\mathsf{\text{d}}y{\parallel }_{\stackrel{̄}{q}}\le {C}_{\gamma ,\tau ,\stackrel{̄}{p},\lambda ,N}\parallel f{\parallel }_{\stackrel{̄}{p}},$
(2.9)

where $0\le \tau , $0\le \gamma and $1∕\stackrel{̄}{p}+\left(\gamma +\tau +\lambda \right)∕N=1+1∕\stackrel{̄}{q}$ with $1/\stackrel{̄}{p}+1/{\stackrel{̄}{p}}^{\prime }=1$.

It follows first from inequality (2.9) and then the Hölder inequality that, for any r > max{(N - ξ)/(N - α), (N - η)/(N - α)},

$\begin{array}{cc}\hfill \parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}\phantom{\rule{1em}{0ex}}& \le \phantom{\rule{1em}{0ex}}C\parallel \underset{{\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}}{\int }\mid x-y{\mid }^{\alpha -N}\mid y{\mid }^{-s}\left[{\stackrel{̄}{v}}_{\lambda }{\left(y\right)}^{q-1}\left({\stackrel{̄}{v}}_{\lambda }\left(y\right)-\stackrel{̄}{v}\left(y\right)\right)\right]\mathsf{\text{d}}y{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}\hfill \\ \le \phantom{\rule{1em}{0ex}}C\parallel {\stackrel{̄}{v}}_{\lambda }{\parallel }_{{L}^{\varphi }\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}^{q-1}\parallel {\stackrel{̄}{v}}_{\lambda }\left(y\right)-\stackrel{̄}{v}\left(y\right){\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)},\hfill \end{array}$
(2.10)

where $\varphi =\frac{q-1}{\frac{\left(N-\alpha \right)q+\xi }{N}-1}.$

Similarly, one can show that

$\parallel {\stackrel{̄}{v}}_{\lambda }-\stackrel{̄}{v}{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}\le C\parallel {ū}_{\lambda }{\parallel }_{{L}^{\beta }\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}^{p-1}\parallel {ū}_{\lambda }\left(y\right)-ū\left(y\right){\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)},$
(2.11)

where $\beta =\frac{p-1}{\frac{\left(N-\alpha \right)p+\eta }{N}-1}$.

Combining (2.10) and (2.11), we arrive at

$\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}\le C\parallel {\stackrel{̄}{v}}_{\lambda }{\parallel }_{{L}^{\varphi }\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}^{q-1}\parallel {ū}_{\lambda }{\parallel }_{{L}^{\beta }\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}^{p-1}\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}.$
(2.12)

By the integrability conditions, we can choose M sufficiently large, such that for λ ≤ -M, we have

$C\parallel {\stackrel{̄}{v}}_{\lambda }{\parallel }_{{L}^{\varphi }\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}^{q-1}\parallel {ū}_{\lambda }{\parallel }_{{L}^{\beta }\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}^{p-1}\le \frac{1}{2}.$
(2.13)

These imply that $\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}=0$. In other words, ${\mathrm{\Sigma }}_{\lambda }^{ū}$ must be measure zero, and hence empty. Similarly, one can show that ${\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}$ is empty. Step 1 is complete.

Step 2. Now we have that for λ ≤ -M,

$ū\left(x\right)\ge {ū}_{\lambda }\left(x\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\stackrel{̄}{v}\left(x\right)\ge {\stackrel{̄}{v}}_{\lambda }\left(x\right),\phantom{\rule{1em}{0ex}}\forall x\in {\sum }_{\lambda }.$
(2.14)

Thus, we can move the plane λ ≤ -M to the right as long as (2.14) holds. Suppose that at one λ 0 < 0, we have, on ${\sum }_{\lambda }$

$ū\left(x\right)\ge {ū}_{{\lambda }_{0}}\left(x\right)\phantom{\rule{1em}{0ex}}\mathsf{\text{and}}\phantom{\rule{1em}{0ex}}\stackrel{̄}{v}\left(x\right)\ge {\stackrel{̄}{v}}_{{\lambda }_{0}}\left(x\right).$

But either

$meas\phantom{\rule{0.3em}{0ex}}\left\{x\in {\sum }_{{\lambda }_{0}}\mid ū\left(x\right)>{ū}_{{\lambda }_{0}}\left(x\right)\right\}>0$

or

$meas\phantom{\rule{0.3em}{0ex}}\left\{x\in {\sum }_{{\lambda }_{0}}\mid \stackrel{̄}{v}\left(x\right)>{\stackrel{̄}{v}}_{{\lambda }_{0}}\left(x\right)\right\}>0.$

Then, we want to show that the plane can be moved further to the right, i.e., there exists an ε depending on N, p, q and the solution ($ū$, $\stackrel{̄}{v}$) such that (2.14) holds on ${\sum }_{\lambda }$ for all λ [λ 0, λ 0 + ε).

Assume that $meas\left\{x\in {\sum }_{{\lambda }_{0}}\mid \stackrel{̄}{v}\left(x\right)>{\stackrel{̄}{v}}_{{\lambda }_{0}}\left(x\right)\right\}>0$. By (2.4), we know that $ū\left(x\right)>{ū}_{{\lambda }_{0}}\left(x\right)$ in the interior of ${\sum }_{{\lambda }_{0}}$. Define $\stackrel{^}{{\sum }_{{\lambda }_{0}}^{ū}}=\left\{x\in {\sum }_{{\lambda }_{0}}\mid ū\left(x\right)\le {ū}_{{\lambda }_{0}}\left(x\right)\right\}$ and $\stackrel{^}{{\sum }_{{\lambda }_{0}}^{\stackrel{̄}{v}}}=\left\{x\in {\sum }_{{\lambda }_{0}}\mid \stackrel{̄}{v}\left(x\right)\le {\stackrel{̄}{v}}_{{\lambda }_{0}}\left(x\right)\right\}$. It is clear that $\stackrel{^}{{\sum }^{{\stackrel{̄}{v}}_{\lambda }}\phantom{\rule{2.77695pt}{0ex}}}$ has measure zero, and ${lim}_{\lambda \to {\lambda }_{0}}{\sum }_{\lambda }^{ū}\subset \stackrel{^}{{\sum }_{{\lambda }_{0}}^{ū}}$ in the sense of measures. The same conclusion holds for $\stackrel{̄}{v}$. Let G* be the reflection of the set G about the plane x 1 = λ. We see from (2.10) and (2.11) that

$\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}\le C\parallel {\stackrel{̄}{v}}_{\lambda }{\parallel }_{{L}^{\varphi }\left({\left({\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}\right)}^{*}\right)}^{q-1}\parallel {ū}_{\lambda }{\parallel }_{{L}^{\beta }\left({\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}^{*}\right)}^{p-1}\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}.$
(2.15)

Again, the integrability of $ū$ and $\stackrel{̄}{v}$ ensures that one can choose ε small enough, such that for all λ [λ 0, λ 0 + ε),

$C\parallel {\stackrel{̄}{v}}_{\lambda }{\parallel }_{{L}^{\varphi }\left({\left({\Sigma }_{\lambda }^{\stackrel{̄}{v}}\right)}^{*}\right)}^{q-1}\parallel {ū}_{\lambda }{\parallel }_{{L}^{\beta }\left({\left({\Sigma }_{\lambda }^{ū}\right)}^{*}\right)}^{p-1}\le \frac{1}{2}.$

Now by (2.15) we have

$\parallel {ū}_{\lambda }-ū{\parallel }_{{L}^{r}\left({\mathrm{\Sigma }}_{\lambda }^{ū}\right)}=0$

and therefore ${\mathrm{\Sigma }}_{\lambda }^{ū}$ is empty. A similar argument shows that ${\mathrm{\Sigma }}_{\lambda }^{\stackrel{̄}{v}}$ is empty too.

Step 3. If the plane stops at x 1 = λ 0 for some λ 0 < 0, then $ū$ and $\stackrel{̄}{v}$ must be symmetric and monotone about the plane x 1 = λ 0. This implies that $ū$ and $\stackrel{̄}{v}$ have no singularity at the origin. But the equations in (2.2) tell us that this is impossible if $ū\left(x\right)$ and $\stackrel{̄}{v}\left(x\right)$ are nontrivial. Hence, we can move the plane to x 1 = 0. Then, $ū\left(x\right)$ and $\stackrel{̄}{v}\left(x\right)$ are symmetric about the plane origin. Then u = v = 0. The proof of Theorem 1.1 is complete.

## 3. Symmetry and monotonicity

In this section, we prove Theorem 1.2 which shows that the non-negative solutions of System (1.1) are radially symmetric and decreasing with respect to some point in N .

Proof of Theorem 1.2. We show that $ū$ and $\stackrel{̄}{v}$ are symmetric with respect to some plane parallel x 1 = 0. Indeed, if λ 0 < 0, such as the steps of Theorem 1.1, we know $ū$ and $\stackrel{̄}{v}$ are symmetric with respect to the hyperplane x 1 = λ 0. If λ 0 = 0, we conclude that ${ū}_{0}\left(x\right)\ge ū\left(x\right)$ and ${\stackrel{̄}{v}}_{0}\left(x\right)\ge \stackrel{̄}{v}\left(x\right)$ for all x Σ0. On the other hand, we perform the moving plane procedure from the right and find a corresponding ${\lambda }_{0}^{r}\ge 0$. If ${\lambda }_{0}^{r}>0$, an analogue to Theorem 1.1 shows that $ū$ and $\stackrel{̄}{v}$ are symmetric with respect to the hyperplane ${x}_{1}={\lambda }_{0}^{r}$. If ${\lambda }_{0}^{r}=0$, we conclude that ${ū}_{0}\left(x\right)\ge ū\left(x\right)$ and ${\stackrel{̄}{v}}_{0}\left(x\right)\ge \stackrel{̄}{v}\left(x\right)$ for all x Σ0. From above we can conclude $ū$ and $\stackrel{̄}{v}$ are symmetric with respect to the plane x 1 = 0. We perform this moving plane procedure taking planes perpendicular to any direction, and for each direction γ N , |γ| = 1, we can find a plane T γ with the property that both $ū$ and $\stackrel{̄}{v}$ are symmetric with respect to T γ . A simple argument shows that all these planes intersect at a single point, or $ū$=$\stackrel{̄}{v}$ = 0. The proof of Theorem 1.2 is complete.

## References

1. Chen WX, Li CM: Classification of positive solutions for nonlinear differential and integral systems with critical exponents. Acta Mathematica Scientia 2009,29(4):949-960. 10.1016/S0252-9602(09)60079-5

2. Chen WX, Li CM, Ou B: Classification of solutions for an integral equation. Comm Pure Appl Math 2006, 59: 330-343. 10.1002/cpa.20116

3. Figueiredo DG, Peral I, Rossi JD: The critical hyperbola for a Hamiltonian elliptic system with weights. Annali Matematica 2008, 187: 531-545. 10.1007/s10231-007-0054-1

4. Clément Ph, Figueiredo DG, Mitidieri E: Positive solutions of semilinear elliptic systems. Comm Partial Diff Equ 1992, 17: 923-940. 10.1080/03605309208820869

5. Peletier LA, Van der Vorst RCAM: Existence and nonexistence of positive solutions of nonlinear elliptic systems and the Biharmonic equation. Diff Integral Equ 1992,5(4):747-767.

6. Figueiredo DG, Felmer PL: On superquadratic elliptic systems. Trans Amer Math Soc 1994, 343: 99-116. 10.2307/2154523

7. Figueiredo DG, Ruf B: Elliptic systems with nonlinearities of arbitrary growth. Mediterr J Math 2004,1(4):417-431. 10.1007/s00009-004-0021-7

8. Felmer PL: Nonexistence and symmetry theorems for elliptic systems in N . Rendiconti Circolo Mate. Palermo, Series II, Tomo 1994,40(3):259-284.

9. Serrin J: A symmetry problem in potential theory. Arch Rational Mech Anal 1971, 43: 304-318.

10. Gidas B, Ni WM, Nirenberg L: Symmetry and related properties via the maximum principle. Comm Math Phys 1979, 68: 209-243. 10.1007/BF01221125

11. Chen WX, Li CM: The best constant in some weighted Hardy-Littlewood-Sobolev inequality. Proc AMS 2008, 136: 955-962.

12. Ma L, Chen DZ: A Liouville-type theorem for an integral system. Comm Pure Appl Anal 2006,5(4):855-859.

13. Huang XT, Li DS, Wang LH: Existence and symmetry of positive solutions of an integral equation system. Math Comp Model 2010, 52: 892-901. 10.1016/j.mcm.2010.05.020

14. Polacik P, Quittner P, Souplet P: Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems. Duke Math J 2007, 139: 555-579. 10.1215/S0012-7094-07-13935-8

15. Hu B: Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition. Diff Integ Equ 1996, 9: 891-901.

16. Chen WX, Li CM: An integral system and the Lane-Emdem conjecture. Disc Cont Dyn Sys 2009,24(4):1167-1184.

17. Figueiredo DG, Felmer PL: A liouvilie-type theorem for systems. Ann Scuola Norm Sup Pisa 1994,21(3):387-397.

18. Zhang ZC, Wang WM, Li KT: Liouville-type theorems for semilinear elliptic systems. J Partial Diff Equ 2005,18(4):304-310.

19. Zhang ZC, Zhu LP: Nonexistence and radial symmetry of positive solutions of semilinear elliptic systems. Disc Dyn Nature Soc 2009, 2009: 8. Article ID 629749

20. Zhu LP, Zhang ZC, Wang WM: On the positive solutions for a class of semilinear elliptic systems. Math Appl 2006,19(2):440-445.

21. Mitidieri E: A Rellich type identity and applications. Comm PDE 1993, 18: 125-151. 10.1080/03605309308820923

22. Mitidieri E: Nonexistence of positive solutions of semilinear elliptic systems in N . Diff Integral Equ 1996,9(3):465-479.

23. Souto MA: Sobre a existência de solucões positivas para sistemas cooperativos nao linears. PhD Thesis, UNICAMP 1992.

24. Serrin J, Zou H: Nonexistence of positive solutions of semi-linear elliptic systems. Discourse Math Appl Texas A&M Univ 1994, 3: 56-69.

25. Van der Vorst RCAM: Variational identities and applications to differential systems. Arch Rational Mech Anal 1991, 116: 375-398.

26. Hulshof J, Van der Vorst RCAM: Differential Systems with strongly indefnite variational structure. J Funct Anal 1993, 114: 32-58. 10.1006/jfan.1993.1062

27. Serrin J, Zou H: The existence of positive entire solutions of elliptic Hamitonian systems. Comm Partial Diff Equ 1998,23(3-4):577-599.

28. Souto MA: A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems. Diff Integral Equ 1995, 8: 1245-1258.

29. Lieb E: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann Math 1983, 118: 349-371. 10.2307/2007032

30. Figueierdo DG: Non-linear elliptic systems. Anais Acad Brasl Cie 2000,72(4):453-469. 10.1590/S0001-37652000000400002

31. Gidas B, Ni WM, Nirenberg L: Symmetry of positive solutions of nonlinear elliptic equations in N . In mathematical analysis and applications. In Part A Adv Math Suppl Stud. Volume 7A. Academic Press, New York; 1981:369-402.

32. Caffarelli L, Gidas B, Spruck J: Asymptotic symmetry and local behviaior of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math 1989, 42: 271-297. 10.1002/cpa.3160420304

33. Chen WX, Li CM: Classification of solutions of some nonlinear elliptic equations. Duke Math J 1991, 63: 615-622. 10.1215/S0012-7094-91-06325-8

34. Li CM: Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent Math 1996, 123: 221-231.

35. Wei JC, Xu XW: Classification of solutions of higher order conformally invariant equations. Math Ann 1999, 313: 207-228. 10.1007/s002080050258

36. Chen SH, Lu GZ: Existence and nonexistence of positive radial solutions for a class of semi-linear elliptic system. Nonlinear Anal 1999,38(7):919-932. 10.1016/S0362-546X(98)00143-6

37. Zhang ZC, Hu B: Gradient blowup rate for a semilinear parabolic equation. Disc Cont Dyn Sys 2010, 26: 767-779.

38. Zhang ZC, Guo ZM: Structure of nontrivial non-negative solutions of singularly perturbed quasilinear Dirichlet problems. Math Nachr 2007,280(13-14):1620-1639. 10.1002/mana.200510568

39. Zhang ZC, Li KT: Radial oscillatory solutions of some quasi-linear elliptic equations. Comp Math Appl 2004,47(8-9):1327-1334. 10.1016/S0898-1221(04)90126-5

40. Chen WX, Li CM, Ou B: Classification of solutions for a system of integral equations. Comm Partial Diff Equ 2005, 30: 59-65. 10.1081/PDE-200044445

41. Chen WX, Li CM, Ou B: Qualitative propertives of solutions for an integral equation. Disc Cont Dyn Sys 2005, 12: 347-354.

42. Stein EM, Weiss G: Fractional integrals in n-dimensional Euclidean space. J Math Mech 1958, 7: 503-514.

## Acknowledgements

We would thank the anonymous referees very much for their valuable corrections and suggestions. This work was supported by Youth Foundation of NSFC (No. 10701061) and Fundamental Research Funds for the Central Universities of China.

## Author information

Authors

### Corresponding author

Correspondence to Zhengce Zhang.

### Competing interests

The author declares that he has no competing interests.

## Rights and permissions

Reprints and Permissions

Zhang, Z. Nonexistence of positive solutions of an integral system with weights. Adv Differ Equ 2011, 61 (2011). https://doi.org/10.1186/1687-1847-2011-61

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1847-2011-61

### Keywords

• integral system
• moving plane method
• nonexistence 