  We consider a general model of a disordered copolymer with adsorption. This
includes, as particular cases, a generalization of the copolymer at a selective
interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9--13], pinning
and wetting models in various dimensions, and the Poland--Scheraga model of DNA
denaturation. We prove a new variational upper bound for the free energy via an
estimation of noninteger moments of the partition function. As an application,
we show that for strong disorder the quenched critical point differs from the
annealed one, for example, if the disorder distribution is Gaussian. In
particular, for pinning models with loop exponent $0<\alpha<1/2$ this implies
the existence of a transition from weak to strong disorder. For the copolymer
model, under a (restrictive) condition on the law of the underlying renewal, we
show that the critical point coincides with the one predicted via
renormalization group arguments in the theoretical physics literature. A
stronger result holds for a "reduced wetting model" introduced by Bodineau and
Giacomin [J. Statist. Phys. 117 (2004) 801--818]: without restrictions on the
law of the underlying renewal, the critical point coincides with the
corresponding renormalization group prediction.
