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Nonlinear elliptic boundary-value problems in unbounded domains. (English) Zbl 0780.35044
From the author’s introduction: “It is well known that in the monotone method for the elliptic boundary value problem \[ -Lu=f(x,u)\text{ in }\Omega, \qquad Bu=g(x,u) \text{ on } \partial\Omega, \] where \(L\) and \(B\) are the elliptic and boundary operators given in the form \[ Lu=\sum^ n_{i,j=1} a_{ij}(x) u_{x_ i x_ j}+\sum^ n_{i=1} b_ i(x)u_{x_ i},\qquad Bu=\alpha \partial u/\partial\nu+\beta u, \] under suitable hypotheses on \(f\) and \(g\), the iteration process \[ \begin{aligned} - &Lu^{(k)}+ cu^{(k)}= cu^{(k-1)}+ f(x,u^{(k-1)}) \quad \text{in }\Omega,\\ &Bu^{(k)}+ bu^{(k)}= bu^{(k-1})+ g(x,u^{(k-1)}) \quad \text{on } \partial\Omega\end{aligned} \] yields a unique monotone sequence \(\{u^{(k)}\}\) which converges monotonically to a solution of the boundary value problem, if \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) and the initial iteration is either an upper solution or a lower solution. (…)
In this paper we extend the above result to the case where \(\Omega\) is a general unbounded domain in \(\mathbb{R}^ n\), including the exterior of a bounded domain and the whole space \(\mathbb{R}^ n\)”.
Reviewer: M.Chicco (Genova)

35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
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