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 $$\begin{cases} -Δ_p u = a(x) u^q & \quad \hbox{ in $Ω$},\\ u >0 & \quad \hbox{ in $Ω$}, \\ u =0 & \quad \hbox{ on $\partial Ω$}, \end{cases}$$ when $Ω\subset \mathbb{R}^N$ 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 $(-Δ)^s_p$, $s\in (0,1)$, $s \approx 1$, 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 $u$.

Theorems & Definitions (68)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4: Exact concavity
• Remark 1.5
• Remark 1.6
• Example 1.7: Hardy-Hénon type equation
• Theorem 1.8: Perturbed concavity I
• Corollary 1.9
• Theorem 1.10: Perturbed concavity II