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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
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