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 $$\{\begin{array}{ll}-\hat{I}{”}_{p}u=a(x)u^q& in\hat{I}\text{c◯},\\ u>0& in\hat{I}\text{c◯},\\ u=0& on\partial \hat{I}\text{c◯},\end{array}$$ when $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ 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 [Kennington, Indiana Univ. Math. J., 1985] and [Sakaguchi, Ann. Sc. Norm. Super. Pisa, 1987]. 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 $(-\hat{I}”{)}_p^s$, $s\in (0,1)$, $s\approx 1$, 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 $u$.