We construct new solutions of ten-dimensional supergravity with sixteen supercharges which describe the backreaction of Dp-branes with spherical worldvolume. These solutions are holographically dual to the (p+1)-dimensional maximally supersymmetric Yang-Mills theory on S^p+1. The finite size of the sphere provides an IR cut-off for the gauge theory which is manifested in the supergravity solution as a smooth cap-off of the geometry. In the UV the size of the sphere plays no role and the

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... nds asymptote to the well-known supergravity solutions that describe the near-horizon limit of flat Dp-branes. We compute the on-shell action of our spherical brane solutions and show that it is in agreement with recent supersymmetric localization results for the free energy of maximal SYM theory on S^p+1.