*Submitted Paper*

**Inserted:** 15 may 2024

**Last Updated:** 15 may 2024

**Pages:** 53

**Year:** 2024

**Abstract:**

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler
equations of the type $$ \begin{cases} -\Delta_{p} u = a(x) u^{q} & \quad \hbox{ in
$\Omega$},\\ u >0 & \quad \hbox{ in $\Omega$}, \\ u =0 & \quad \hbox{ on
$\partial \Omega$}, \end{cases} $$ 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