A Hardy-Hénon equation in R^N with sublinear absorption

Abstract

Consider m > 1, N ≥ 1 and max{−2, −N} <σ< 0. The Hardy-Hénon equation with sublinear absorption −v(x) − |x| σ v(x) + 1 m − 1 v1/m(x) = 0, x ∈ RN, is shown to have at least one solution v ∈ H1(RN ) ∩ L(m+1)/m(RN ), which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity v ∈ W2,q (RN )for q ∈ [1, N/|σ|). A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile. A byproduct of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient.