Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 16, pp. 1-10.

Title: Nonexitence of nontrivial solutions to Dirichlet problems for the fractional Laplacian
       
Authors: Jose Carmona (Univ. de Almeria, Spain)
         Alexis Molino (Univ. de Almeria, Spain)
         
Abstract:
In this article we prove that there are no nontrivial solutions to
the Dirichlet problem for the fractional Laplacian
$$\displaylines{
(-\Delta)^s u =f(u) \quad \text{in }\Omega,\cr
u=0 \quad \text{in } \mathbb{R}^N \backslash \Omega,
}$$
where $\Omega \subset \mathbb{R}^N$ (N≥ 1) is a bounded domain,
and f is locally Lipschitz with non-positive primitive
$F(t)= \int_0^t f(\tau)d\tau$.

Submitted March 30, 2022 Published February 17, 2023.
Math Subject Classifications: 35J05, 35J15, 35J25.
Key Words: Fractional Laplacian; Dirichlet problem; nonexistence of solutions.