For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a “close” and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach’s fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

中文翻译：

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半线性椭圆边值问题的计算机辅助证明

对于有界或无界域上的二阶半线性椭圆边值问题，提出了一种通用计算机辅助方法，用于证明近似解的“紧密”和显式邻域中解的存在性，通过数值方法计算。为了实现这样的存在性和封闭性结果，我们将巴拿赫不动点定理应用于误差的等价问题，即精确解与近似解之间的差异。为定点参数提出的条件的验证需要各种分析和数值技术，例如在近似解处计算线性化的特征值界限。该方法用于证明某些特定例子的存在性和多重性结果。