Weighted Trace Inequalities for Fractional Integrals and Applications to Semilinear Equations

We show that the two-weight trace inequality for the Riesz potentials Iα, ||Iαf||Lp(w) ≤ C|| f||Lp(v), holds if Iαw ∈ Lp′loc (σ) and Iα[(Iαw)p′ σ] ≤ CIαw a.e. Here w and v are non-negative weight functions on Rn, and σ = v1-p′. The converse is also also true under some mild restrictions on w and v. We also consider more general inequalities for measures which are not necessarily absolutely continuous with respect to Lebesgue measure. In contrast to the known characterizations of the trace inequality, this "pointwise" condition is stated explicitly in terms of potentials of w and σ, rather than measures of some subsets of Rn. Applications to the problem of the existence of positive solutions for the semilinear elliptic equation −Δu = σ(x) uq + w(x) (1 < q < ∈) on Rn are given.