## Remark on some conformally invariant integral equations: the method of moving spheres.(English)Zbl 1075.45006

The author uses the so-called method of moving spheres to study some integral equations on $$\mathbb{R}^n$$. These includes the equation $u(x)=\int_{\mathbb{R}^n} {u(y)^{\mu}\over{| x-y| ^{n-\alpha}}}\, dy, \;\;\; x\in \mathbb{R}^n,$ with $$n\in \mathbb{N}$$, $$0<\alpha<n$$, $$\mu>0$$, and the unknown $$u$$ is supposed to be a positive Lebesgue measurable function in $$\mathbb{R}^n$$.
In the case $$\mu={{n+\alpha}\over{n-\alpha}}$$ the above equation is conformally invariant in the following sense: for $$x\in \mathbb{R}^n$$ and $$\lambda>0$$, define $v_{x,\lambda}(\xi)=\left( \lambda\over {| \xi-x| } \right)^{n-\alpha} v(\xi^{x,\lambda}), \;\:\; \xi\in \mathbb{R}^n,$ where $\xi^{x,\lambda}=x+{{\lambda^2 (\xi-x)}\over{| \xi-x| ^2}},$ then if $$u$$ is a solution of the integral equation so is $$u_{x,\lambda}$$ for any $$x\in \mathbb{R}^n$$ and $$\lambda>0$$. It is proved that the positive solutions $$u\in L^\infty_{\text{loc}} (\mathbb{R}^n)$$ of the above equation are of the form $u (x)\equiv \left({a\over {d+| x-\tilde{x}| ^2}} \right)^{(n-\alpha)/2}$ (Theorem 1.1), and that, if $$u\in L^{2n/(n-\alpha)}_{\text{loc}} (\mathbb{R}^n)$$ is a positive solution of the equation, then $$u\in C^\infty (\mathbb{R}^n)$$ (Theorem 1.2).
In the case $$\mu\neq {{n+\alpha}\over{n-\alpha}}$$, the equation is not conformally invariant in the above sense. It is proved (Theorem 1.4) that: (i) For $$0<\mu<n/(n-\alpha)$$, the equation does not have any positive solution $$u$$, unless $$u\equiv \infty$$; (ii) For $$n/(n-\alpha)\leq \mu <(n+\alpha)/(n-\alpha)$$, the equation does not have any positive solution $$u\in L^{n(\mu-1)/\alpha}_{\text{loc}} (\mathbb{R}^n)$$.
The integral equation $u(x)=\int_{\mathbb{R}^n} {{| x-y| ^{p} u(y)^{-q}}}\, dy, \;\;\; x\in \mathbb{R}^n,$ for $$n\in \mathbb{N}$$ and $$p,q>0$$ is also considered. It is proved (Theorem 1.5) that, if $$0<q\leq 1+2n/p$$ and $$u$$ is a nonnegative Lebesgue measurable function in $$\mathbb{R}^n$$ satisfying the above integral equation, then $$q=1+2n/p$$ and for some constants $$a,d>0$$ and some $$\tilde{x}\in \mathbb{R}^n$$, $u (x)\equiv \left({{d+| x-\tilde{x}| ^2}\over a} \right)^{p/2} .$

### MSC:

 45G05 Singular nonlinear integral equations 45M20 Positive solutions of integral equations 35B33 Critical exponents in context of PDEs 35J60 Nonlinear elliptic equations

### Keywords:

method of moving spheres; positive solutions
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### References:

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