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.


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