Concavity and perturbed concavity for p-Laplace equations

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type {−Δpu=a(x)uq in Ω,u>0 in Ω,u=0 on ∂Ω, when Ω⊂RN is a convex domain. In particular, in the subhomogeneous case q∈[0,p−1], the solution u inherits concavity properties from a whenever assumed, while it is proved to be concave up to an error if a is near to a constant. More general problems are also taken into account, including a wider class of nonlinearities. These results generalize some contained in [91] and [120]. Additionally, some results for the singular case q∈[−1,0) and the superhomogeneous case q>p−1, q≈p−1 are obtained. Some properties for the p-fractional Laplacian (−Δ)ps, s∈(0,1), s≈1, are shown as well. We highlight that some results are new even in the semilinear framework p=2; in some of these cases, we deduce also uniqueness (and nondegeneracy) of the critical point of u.