One of the archetypical aggregation-diffusion models is the so-called classical parabolic- elliptic Patlak-Keller-Segel (PKS for short) model. This model was classically introduced as the simplest description for chemotactic bacteria movement in which linear diffusion tendency to spread fights the attraction due to the logarithmic kernel interaction in two dimensions. For this model there is a well-defined critical mass. In fact, here a clear dichotomy arises: if the total mass of the system is less than the critical mass, then the long time asymptotics are described by a self-similar solution, while for a mass larger than the critical one, there is finite time blow-up. In this talk we will first give an overview about some results obtained in the papers [1]-[2] concerning the characterization of the stationary states for a nonlinear variant of the PKS model, of the form (1) ∂tρ = ∆ρm + ∇ · (ρ∇(W ∗ ρ)), being W ∈ C1(Rd \ {0}) a Riesz kernel aggregation, namely W(x) = cd,s|x|2s−d for s ∈ (0, d/2), in the assumptions of dominated diffusion, i.e. when i.e. for m > 2 − (2s)/d. In particular, all stationary states of the model are shown to be radially symmetric decreasing and uniquely identified with global minimizers of the associated free energy functionals. In the second part of the talk we will discuss the recent results established in the joint paper [3], in which an addition of a quadratic diffusion term in equation (1) produces a more precise competition with the aggregation term for small s, as they have the same scaling if s = 0. We characterize the asymptotic behavior of the stationary states behavior as s goes to zero. Finally, we establish the existence of gradient flow solutions to the evolution problem by applying the JKO scheme. References [1] J. A. Carrillo, F. Hoffmann, E. Mainini, B. Volzone. Ground States in the Diffusion-Dominated Regime, Calc. Var. Partial Differ. Equ. 57, No. 5, Paper No. 127, 28 p. (2018). [2] H. Chan, M. Gonz´alez, Y. Huang, E. Mainini, B. Volzone. Uniqueness of entire ground states for the fractional plasma problem., Calc. Var. Partial Differ. Equ. 59, No. 6, Paper No. 195, 41 p. (2020). [3] Y. Huang, E. Mainini, J. L. V´azquez, B. Volzone. Nonlinear aggregation-diffusion equations with Riesz potentials, arXiv:2205.13520 [math.AP] (2022)