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

An existence theory for superposition operators of mixed order subject to jumping nonlinearities