Concavity and perturbed concavity for $p$-Laplace equations
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arXiv
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| Natura: | Preprint |
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2024
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| author | Gallo, Marco Squassina, Marco |
| author_facet | Gallo, Marco Squassina, Marco |
| contents | 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 $u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concavity and perturbed concavity for $p$-Laplace equations Gallo, Marco Squassina, Marco Analysis of PDEs 26B25, 35B09, 35B20, 35B99, 35D30, 35E10, 35J60, 35J62, 35R11 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 $u$. |
| title | Concavity and perturbed concavity for $p$-Laplace equations |
| topic | Analysis of PDEs 26B25, 35B09, 35B20, 35B99, 35D30, 35E10, 35J60, 35J62, 35R11 |
| url | https://arxiv.org/abs/2405.05404 |