  We consider a superposition operator of the form $$ \int_{[0, 1]} (-\Delta)^s
u\, d\mu(s),$$ for a signed measure $\mu$ on the interval of fractional
exponents $[0,1]$, joined to a nonlinearity whose term of homogeneity equal to
one is "jumping", i.e. it may present different coefficients in front of the
negative and positive parts. The signed measure is supposed to possess a
positive contribution coming from the higher exponents that overcomes its
negative contribution (if any). The problem taken into account is also of
"critical" type, though in this case the critical exponent needs to be
carefully selected in terms of the signed measure $\mu$. Not only the operator
and the nonlinearity considered here are very general, but our results are new
even in special cases of interest and include known results as particular
subcases. The possibility of considering operators "with the wrong sign" is
also a complete novelty in this setting.
