We study the compatibility between the PBEsol exchange-correlation energy functional of Phys. Rev. Lett. 100, 136406 (2008) and the rVV10 van der Waals nonlocal correlation functional of Phys. Rev. B 87, 041108 (2013). By applying a density-gradient dependence in the expression of rVV10, we develop a new functional: The PBEsol + rVV10s functional, which is remarkably accurate for layered solids, rare-gas crystals, densely packed bulk solids, and the adsorption of noble-gas atoms on metal surfaces, and is also competitive for noncovalent interactions between molecular complexes as well as for potential-energy curves of noble-gas dimers. The capacity of this dispersion-corrected functional to describe various interactions in solids makes it useful for electronic-structure calculations in solid-state physics and materials science. Peer reviewed