Nonexitence of nontrivial solutions to Dirichlet problems for the fractional Laplacian

[摘要] In this article we prove that there are no nontrivial solutions to the Dirichlet problem for the fractional Laplacian $$ \displaylines{(-\Delta)^s u =f(u) \quad \text{in }\Omega,\\ u=0 \quad \text{in } \mathbb{R}^N \backslash \Omega,}$$ where\(\Omega \subset \mathbb{R}^N\) (\(N\geq 1\)) is a bounded domain, and f is locally Lipschitz with non-positive primitive \(F(t)= \int_0^t f(\tau)d\tau\).

[学科分类] 数学（综合）

[关键词] Fractional Laplacian;Dirichlet problem;nonexistence of solutions