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On a singular nonlinear elliptic problem. (English) Zbl 0614.35037

This paper is concerned with the elliptic boundary value problem of the form \[ Lu(x)=-\sum^{n}_{i,j=1}(\partial /\partial x_ i)(a_{ij}(\partial /\partial x_ j)u(x))=f(x,u(x)),\quad for\quad x\in \Omega;\quad u(x)=0\quad for\quad x\in \partial \Omega \] where \(\Omega\) is a bounded region in \(R^ n\), \(n\geq 3\), \(\partial \Omega\) is the boundary of \(\Omega\), k(x) is a nonnegative measurable real function, and the operator L is uniformly elliptic. First, various inequalities for G(x,s) are proved, where G(x,s) is the Green’s function for the Dirichlet problem associated to L in \(\Omega\). Then, using these inequalities, a study is made of the action of the integral operator defined by the kernel G on unbounded functions with a prescribed growth near the boundary. Finally, under appropriate assumptions, the author proves the existence and uniqueness of a solution \(u\geq 0\) in \(C^ 1({\bar \Omega})\) for the equivalent fixed point problem \(u(x)=\int_{\Omega}G(x,s)k(s)[u(s)]^{-\alpha} ds\).
Reviewer: Xu Zhenyuan

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35C15 Integral representations of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
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