Nonexitence of nontrivial solutions to Dirichlet problems for the fractional Laplacian

Abstract

In this article we prove that there are no nontrivial solutions tothe 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 \(F(t)= \int_0^t f(\tau)d\tau\).