We consider one-sided subshifts σ with some potential functions φ which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures ν and invariant measures μ absolutely continuous with respect to ν, where μ may be finite or infinite. We show that the systems (σ, μ) are exact, and μ are weak Gibbs measures and equilibriums for φ. We also discuss uniqueness of equilibriums and phase transition. These results can be applied to some expanding dynamical systems with an indifferent fixed point.