Calculus of Variations and Geometric Measure Theory

M. Gallo - M. Squassina

Concavity and perturbed concavity for $p$-Laplace equations

created by gallo on 15 May 2024


Submitted Paper

Inserted: 15 may 2024
Last Updated: 15 may 2024

Pages: 53
Year: 2024

ArXiv: 2405.05404 PDF
Links: Researchgate


In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type $$ \begin{cases} -\Deltap u = a(x) uq & \quad \hbox{ in $\Omega$},\\ u >0 & \quad \hbox{ in $\Omega$}, \\ u =0 & \quad \hbox{ on $\partial \Omega$}, \end{cases} $$ when $\Omega \subset \mathbb{R}^N$ 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 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 $q \in [-1,0)$ and the superhomogeneous case $q>p-1$, $q \approx p-1$ are obtained. Some properties for the $p$-fractional Laplacian $(-\Delta)^s_p$, $s\in (0,1)$, $s \approx 1$, 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 $u$.

Keywords: p-Laplacian, Concavity of solutions, Perturbation results, Quasilinear equation, Nonautonomous equations, Fractional equation, Uniqueness of critical point, Nondegenerate critical point