Nanopores in solid state membranes are a tool able to probe nanofluidic phenomena or can act as a single molecular sensor. They also have diverse applications in filtration, desalination, or osmotic power generation. Many of these applications involve chemical, or hydrostatic pressure differences which act on both the supporting membrane, and the ion transport through the pore. By using pressure differences between the sides of the membrane and an alternating current approach to probe ion transport, we investigate two distinct physical phenomena: the elastic deformation of the membrane through the measurement of strain at the nanopore, and the growth of ionic current rectification with pressure due to pore entrance effects. These measurements are a significant step toward the understanding of the role of elastic membrane deformation or fluid flow on linear and nonlinear transport properties of nanopores.