We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion,We consider data f ∈ L 1 (R N ) and all exponents 0 < σ < 2 and m > 0. Existence and uniqueness of a weak solution is established for m > m * = (N − σ) + /N , giving rise to an L 1 -contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range 0 < m ≤ m * existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above m * . We also study the dependence of solutions on f, m and σ. Moreover, we consider the above questions for the problem posed in a bounded domain.2000 Mathematics Subject Classification. 26A33, 35A05, 35K55, 76S05

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We develop a theory of existence, uniqueness and regularity for the following porous medium equation with fractional diffusion,with m > m * = (N − 1)/N , N ≥ 1 and f ∈ L 1 (R N ). An L 1 -contraction semigroup is constructed and the continuous dependence on data and exponent is established. Nonnegative solutions are proved to be continuous and strictly positive for all x ∈ R N , t > 0.

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We study the regularity properties of the solutions to the nonlinear equation with fractional diffusionIf the nonlinearity satisfies some not very restrictive conditions: ϕ ∈ C 1,γ (R), 1 + γ > σ, and ϕ ′ (u) > 0 for every u ∈ R, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even C ∞ regularity. Degenerate and singular cases, including the power nonlinearity ϕ(u) = |u| m−1 u, m > 0, are also considered, and the existence of classical solutions in the power case is proved.

We study the nonlinear and nonlocal Cauchy problemwhere L is a Lévy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity ϕ is nondecreasing and continuous, and the initial datum u 0 is assumed to be in L 1 (R N ). We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, ϕ(u) = |u| m−1 u, m > 1, these solutions turn out to be bounded and Hölder continuous for t > 0. We also describe the large time behaviour when the nonlinearity resembles a power for u ≈ 0 and the kernel associated to L is close at infinity to that of the fractional Laplacian.

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