  We study the limiting behavior of solutions to boundary value nonlinear
problems involving the fractional Laplacian of order $2s$ when the parameter
$s$ tends to zero. In particular, we show that least-energy solutions converge
(up to a subsequence) to a nontrivial nonnegative least-energy solution of a
limiting problem in terms of the logarithmic Laplacian, i.e., the
pseudodifferential operator with Fourier symbol $\ln(|\xi|^2)$. These results
are motivated by some applications of nonlocal models where a small value for
the parameter $s$ yields the optimal choice. Our approach is based on
variational methods, uniform energy-derived estimates, and the use of a new
logarithmic-type Sobolev inequality.
