Concavity and perturbed concavity for p-Laplace equations

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type cases -Δₚ u = a(x) uq & inΩ,\\ u >0 & inΩ, \\ u =0 & on∂ Ω, cases 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 cases are also taken into account, including a wider class of nonlinearities. These results generalize some contained in [Kennington, Indiana Univ. Math. J., 1985] and [Sakaguchi, Ann. Sc. Norm. Super. Pisa, 1987]. 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)ˢₚ, s∈ (0,1), s ≈ 1, are shown as well. We highlight that some results are new even in the semilinear case $p=2$; in some of these cases, we deduce also uniqueness (and nondegeneracy) of the critical point of u.