This paper presents Tikhonov- and iterated soft-shrinkage regularization methods for nonlinear inverse medium scattering problems. Motivated by recent sparsity-promoting reconstruction schemes for inverse problems, we assume that the contrast of the medium is supported within a small subdomain of a known search domain and minimize Tikhonov functionals with sparsity-promoting penalty terms based on Lp-norms. Analytically, this is based on scattering theory for the Helmholtz equation with the refractive index in Lp, 1 <p <∞, and on crucial continuity and compactness properties of the contrast-to-measurement operator. Algorithmically, we use an iterated soft-shrinkage scheme combined with the differentiability of the forward operator in Lp to approximate the minimizer of the Tikhonov functional. The feasibility of this approach together with the quality of the obtained reconstructions is demonstrated via numerical examples.