## Characterization of balls in terms of Bessel-potential integral equation.(English)Zbl 1229.31005

Summary: For a bounded $$C^{1}$$ domain $$\Omega \subset \mathbb R^N$$, we consider the Bessel potential
$u(x)=\int_\Omega g_\alpha^{}(x-y)\, dy$ for $$2\leq \alpha <N$$. We show that $$u=\text{constant}$$ on $$\partial \Omega$$ if and only if $$\Omega$$ is a ball. The more general Bessel-potential integral equation
$u(x)=\int_\Omega g^{}_\alpha(x-y)\, h\big(u(y)\big)\, dy$ is also studied. A similar characterization of balls holds under certain assumptions on $$u$$ and $$h\big(u(y)\big)$$. To establish our main results, we use an integral form of the celebrated moving plane method of A. D. Alexandroff [Ann. Mat. Pura Appl., IV. Ser. 58, 303–315 (1962; Zbl 0107.15603)], J. Serrin [Arch. Ration. Mech. Anal. 43, 304–318 (1971; Zbl 0222.31007)], and B. Gidas, W.-M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209–243 (1979; Zbl 0425.35020); Adv. Math., Suppl. Stud. 7A, 369-402 (1981; Zbl 0469.35052)] developed by W. Chen, C. Li and B. Ou [Commun. Pure Appl. Math. 59, No. 3, 330–343 (2006; Zbl 1093.45001); corrigendum 59, No. 7, 1064 (2006; Zbl 1093.45001)].

### MSC:

 31B10 Integral representations, integral operators, integral equations methods in higher dimensions 35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
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### References:

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