This talk is concerned with the inverse acoustic scattering problem for detecting a bounded penetrable obstacle with a conductivity boundary condition, which may contain several unknown embedded objects inside. We proved that the penetrable obstacle can be uniquely recovered from the fixed frequency far-field data, ignoring its contents, by reducing the scattering problem into a boundary value problem with L^p data and illustrating in detail the singularity of the solutions corresponding to the points sources and hypersingular sources.