We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.

中文翻译：

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类拉普拉斯分数算子的狄利克雷问题的 Ambrosetti-Prodi 类型结果

我们为分数拉普拉斯和可比算子的非线性狄利克雷问题的粘度和经典解建立 Ambrosetti-Prodi 类型结果。在选择非线性时，我们分别考虑半线性和超线性增长情况。我们使用基于功能集成的方法开发了一种新技术，该方法在非本地环境中比纯分析处理更强大。