Concavity and perturbed concavity for p-Laplace equations

Abstract

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type$\{\begin{array}{cc}-{\Delta }_{p}u=a\left(x\right)u^q& \text{ in Ω},\\ u>0& \text{ in Ω},\\ u=0& \text{ on ∂Ω},\end{array}$ when $\Omega \subset R^N$ is a convex domain. In particular, in the subhomogeneous case $q\in [0,p-1]$, 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 $q\in [-1,0)$ and the superhomogeneous case $q>p-1$, $q\approx p-1$ are obtained. Some properties for the p-fractional Laplacian ${(-\Delta )}_p^s$, $s\in (0,1)$, $s\approx 1$, 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.