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In this note, we will study the problemwhere < s < , (−∆) s p is the nonlocal p-Laplacian de ned below, Ω is a smooth bounded domain. The main point studied in this work is to prove, adapting the techniques used in [ ] for the case p = to the general case p ∈ ( , +∞), the summability of the nite energy solutions in terms of the summability of a source term f(x). The aim of this note is to present the results in a way as elementary as possible.

We prove a quantitative version of the Faber-Krahn inequality for the first eigenvalue of the fractional Dirichlet-Laplacian of order s. This is done by using the so-called Caffarelli-Silvestre extension and adapting to the nonlocal setting a trick by Hansen and Nadirashvili. The relevant stability estimate comes with an explicit constant, which is stable as the fractional order of differentiability goes to 1.2010 Mathematics Subject Classification. 47A75, 49Q20, 35R11.

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