We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely,The problem is posed in {x ∈ R N , t ∈ R} with nonnegative initial data u(x, 0) that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. Here we establish the boundedness and C α regularity of such weak solutions

In this work we consider the problemswhere L is a nonlocal differential operator and Ω is a bounded domain in R N , with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f . In the process we establish an L p -theory, for p 1, associated to these problems and we prove some useful inequalities for the applications.2010 Mathematics Subject Classification. 45K05, 47G20, 35R09, 35D30, 35D35.

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In this work we study the following fractional critical problemwhere Ω ⊂ R n is a regular bounded domain, λ > 0, 0 < s < 1 and n > 2s. Here (−∆) s denotes the fractional Laplace operator defined, up to a normalization factor, byOur main results show the existence and multiplicity of solutions to problem (P λ ) for different values of λ. The dependency on this parameter changes according to whether we consider the concave power case (0 < q < 1) or the convex power case (1 < q < 2 * s −1). These two cases will be treated separately.