Axial symmetry and regularity of solutions to an integral equation in a half-space. (English) Zbl 1239.45005

Lu, Guozhen; Zhu, Jiuyi

Pac. J. Math. 253, No. 2, 455-473 (2011).

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.

Reviewer: Marius Ghergu (Dublin)

MSC:

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

Keywords:

integral equations in half-space; Green function for the polyharmonic operator; moving plane method; lifting method; systems of integral equations