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Axial symmetry and regularity of solutions to an integral equation in a half-space. (English) Zbl 1239.45005
This paper is concerned with qualitative properties of solutions corresponding to the integral equation $u(x)=\int_{\mathbb R^n_+}G(x,y)f(u(y))dy.$ Here $$\mathbb R^n_+=\{x\in \mathbb R^n:x_n>0\}$$, $$G(\cdot,\cdot)$$ is the Green function of $$(-\Delta)^m$$, $$n>2m$$ subject to a Dirichlet boundary condition and $$f:[0,\infty)\rightarrow [0,\infty)$$ is increasing.
The first result of the paper establishes that if $$u\in L^r(\mathbb R^n_+)$$, $$r>n/(n-2m)$$ and one of the following conditions hold:
(i) $$|\partial f(u)/\partial u|\leq c(|u|^{\beta_1}+|u|^{\beta_2})$$ where $$C>0$$, $$\beta_1\geq 0\geq \beta_2$$, $$u^{\beta_1}, u^{\beta_2}\in L^{n/(2m)}(\mathbb R^n_+)$$
(ii) $$f'$$ is nondecreasing and $$f'(u)\in L^{n/(2m)}(\mathbb R^n_+)$$
then $$u$$ is the trivial solution.
The authors also obtain a corresponding result for systems of integral equations. Next, the following integral equation is considered $u(x)=\int_{\mathbb R^n_+}G(x,y)|u|^{p-1}u dy.$ The authors prove that any solution $$u$$ satisfying $$u\in L^{(p-1)n/(2m)}(\mathbb R^n_+)$$, $$p>n/(n-2m)$$ is bounded in $$\mathbb R^n_+$$ and bounded in any $$L^s(\mathbb R^n_+)$$ for all $$s>n/(n-2m)$$. Furthermore, if $$1<p<(n+2m)/(n-2m)$$, then any nonnegative solution $$u$$ such that $$u \in L^{2n/(n-2m)}(\mathbb R^n_+)$$ depends only on the $$x_n$$-variable. The proofs rely on the method of moving planes in integral form combined with lifting method and properties of the Green function for the polyharmonic operator.

##### MSC:
 45G15 Systems of nonlinear integral equations 35J08 Green’s functions for elliptic equations
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