  We prove nonexistence of nontrivial bounded solutions to some nonlinear
problems involving nonlocal operators of the form \[Lu(x)=\sum
a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] These operators
are infinitesimal generators of symmetric L\'evy processes. Our results apply
to even kernels $K$ satisfying that $K(y)|y|^{n+\sigma}$ is nondecreasing along
rays from the origin, for some $\sigma\in(0,2)$ in case $a_{ij}\equiv0$ and for
$\sigma=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix.
  Our nonexistence results concern Dirichlet problems for $L$ in star-shaped
domains with critical and supercritical nonlinearities (where the criticality
condition is in relation to $n$ and $\sigma$).
  We also establish nonexistence of bounded solutions to semilinear equations
involving other nonlocal operators such as the higher order fractional
Laplacian $(-\Delta)^s$ (here $s>1$) or the fractional $p$-Laplacian. All these
nonexistence results follow from a general variational inequality in the spirit
of a classical identity by Pucci and Serrin.
