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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