The Earth’s hydrogen exosphere Lyman-α radiation was mapped with the SWAN/SOHO instrument in January 1996, 1997 and 1998 (low solar activity). The use of a hydrogen absorption cell allowed to disentangle the interplanetary emission from the geocoronal one and to assign the absorbed signal almost entirely to the geocorona. The geocorona was found to extend at least up to 100 Earth Radii (R E ) with an intensity of 5 Rayleigh, an unprecedented distance well exceeding the recent results of LAICA imager (∼50 R E ), and encompassing the orbit of the Moon (∼60 R E ). We developed a numerical kinetic model of the hydrogen atoms distribution in the exosphere which includes the solar Lyman-α radiation pressure and the ionization. The radiation pressure compresses the H exosphere on the day side, producing a bulge of H density between 3 and 20 R E which fits observed intensities very well. The SWAN Lyman-α distribution of intensity was compared both to LAICA (2015) and to OGO-5 (1968) measurements. Integrated H densities of SWAN at a tangent distance of 7 R E are larger than LAICA/OGO-5 by factors 1.1-2.5, while we should expect a stronger effect of the radiation pressure at solar max. We discuss the possible role of H atoms in satellite orbits to explain this apparent contradiction. An onion-peeling technique is used to retrieve hydrogen number density in the exosphere for the three SWAN observations. They show an excess of density versus models at large distances, which is likely due to non-thermal atoms (not in the model).