## Classification of solutions for a system of integral equations.(English)Zbl 1073.45005

Consider the following system of integral equations in $$\mathbb{R}^n$$: $u(x)= \int_{\mathbb{R}^n}|x-y|^{\alpha- n}v^q(y)\,dy,\quad v(x)= \int_{\mathbb{R}^n}|x-y|^{\alpha-n} u^p(y)\,dy,\tag{1}$ where $$0<\alpha< n$$ and $$1/(p+1)+ 1(q+1)= (n- \alpha)/n$$. Assume that $$u\in L^{p+1}(\mathbb{R}^n)$$ and $$v\in L^{q+1}(\mathbb{R}^n)$$. The authors show that if $$u$$ and $$v$$ are solutions of the system (1), then they are radially symmetric and decreasing about some point $$x_0$$.

### MSC:

 45G15 Systems of nonlinear integral equations 35J60 Nonlinear elliptic equations
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### References:

 [1] DOI: 10.2307/2946638 · Zbl 0826.58042 [2] DOI: 10.1007/BF01244896 · Zbl 0784.35025 [3] Brezis H., J. Math. Pure Appl. 58 pp 137– (1979) [4] DOI: 10.1007/BF01217349 · Zbl 0579.35025 [5] DOI: 10.1002/cpa.3160420304 · Zbl 0702.35085 [6] Chang A., Math. Res. Letters 4 pp 1– (1997) · Zbl 0903.53027 [7] DOI: 10.1215/S0012-7094-91-06325-8 · Zbl 0768.35025 [8] DOI: 10.2307/2951844 · Zbl 0877.35036 [9] DOI: 10.1017/CBO9780511569203 [10] Gidas B., which is Vol. 7a of the book series, in: Mathematical Analysis and Applications (1981) · Zbl 0465.35003 [11] Li C., Invent. Math. 123 pp 221– (1996) [12] Li , Y. Remarks on some conformally invariant integral equations: The method of moving spheres . Preprint . [13] DOI: 10.2307/2007032 · Zbl 0527.42011 [14] Lieb E., 2d edition, in: Analysis (2001) · Zbl 0966.26002 [15] Ou B., Houston J. Math. 25 pp 181– (1999) [16] DOI: 10.1007/BF00250468 · Zbl 0222.31007 [17] Stein E., Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501 [18] DOI: 10.1007/s002080050258 · Zbl 0940.35082
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