Searcharxiv⌕ Search

arXiv subjects

Luigi Montoro

Publications and source records attributed to Luigi Montoro.

At least 19 recordsLinked to original sources
arXiv2026

Monotonicity of non-negative solutions of quasilinear elliptic equations in a cylindrical domain

We consider weak solutions to $p$-Laplace equations in cylindrical domains under mixed homogeneous Dirichlet-Neumann boundary conditions. We assume that the right-hand side is positive and locally Lipschitz continuous and we prove that any positive solution is monotone increasing in the $x_N$ direction for any $p>1$. As an application we prove that solutions to Allen-Cahn type equations are one-dimensional as well as a Liouville type result for Lane-Emden type equations.

math.AP↗
arXiv2025

An extension of Cabré-Chanillo theorem to the $p$-laplacian

In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\\ u>0&in\ \Om,\\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment.

math.AP↗
arXiv2024

On mixed local-nonlocal problems with Hardy potential

In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-Δu + (-Δ)^s u - γ\frac{u}{|x|^2}=f \text{ in } Ω, \ u=0 \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $γ> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $γ$.

math.AP↗
arXiv2024

Regularity and symmetry results for the vectorial p-Laplacian

We obtain some regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$}\,.$$ More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.

math.AP↗
arXiv2023

Asymptotic behaviour of solutions to the anisotropic doubly critical equation

The aim of this paper is to deal with the anisotropic doubly critical equation $$-Δ_p^H u - \fracγ{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1} \qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm, $H^\circ$ is the dual norm, $1<p<N$, $0 \leq γ< \left((N-p)/p\right)^p$ and $p^*=Np/(N-p)$. In particular, we provide a complete asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.

math.AP↗
arXiv2023

Optimal second order boundary regularity for solutions to $p$-Laplace equations

Solutions to $p$-Laplace equations are not, in general, of class $C^2$. The study of Sobolev regularity of the second derivatives is, therefore, a crucial issue. An important contribution by Cianchi and Maz'ya shows that, if the source term is in $L^2$, then the field $|\nabla u|^{p-2}\nabla u$ is in $W^{1,2}$. The $L^2$-regularity of the source term is also a necessary condition. Here, under suitable assumptions, we obtain sharp second order estimates, thus proving the optimal regularity of the vector field $|\nabla u|^{p-2}\nabla u$, up to the boundary.

math.AP↗
arXiv2022

On the Pohozaev identity for quasilinear Finsler anisotropic equations

In this paper we derive the Pohozaev identity for quasilinear equations \begin{equation}\tag{$E$}\label{eq:p} -\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))=g(x, u) \quad \text {in}\,\, Ω, \end{equation} involving the anisotropic Finsler operator $-\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$. In particular, by means of fine regularity results on the vectorial field $B'(H(\nabla u))\nabla H(\nabla u)$, we prove the identity for weak solutions and in a direct way.

math.AP↗
arXiv2021

Symmetry and monotonicity results for solutions of vectorial $p$-Stokes systems

In this paper we shall study qualitative properties of a $p$-Stokes type system, namely $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u})\,\, \mbox{ in $Ω$},$$ where ${\boldsymbol Δ}_p$ is the $p$-Laplacian vectorial operator. More precisely, under suitable assumptions on the domain $Ω$ and the function $\boldsymbol{ f}$, it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators.

math.AP↗
arXiv2021

Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.

math.AP↗
arXiv2021

Fractional Hardy equations with critical and supercritical exponents

We study the existence/nonexistence and qualitative properties of the positive solutions to the problem \begin{align*} (-Δ)^s u -θ\frac{u}{|x|^{2s}}&=u^p - u^q \quad\text{in }\,\, \mathbb{R}^N,\quad u > 0 \quad\text{in }\,\, \mathbb{R}^N, \quad u \in \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), \end{align*} where $s\in (0,1)$, $N>2s$, $q>p\geq{(N+2s)}/{(N-2s)}$, $θ\in(0, Λ_{N,s})$ and $Λ_{N,s}$ is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions we mean, both the radial symmetry that is obtained by using the moving plane method in a nonlocal setting on the whole $\mathbb{R}^N$, and upper bound behavior of the solutions. To this last end we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.

math.AP↗
arXiv2019

Integral representation using Green function for fractional Hardy equation

Our main aim is to study Green function for the fractional Hardy operator $P:=(-Δ)^s -\fracθ{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<θ<Λ_{N,s}$ and $Λ_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

math.AP↗