arXiv subjects

Francesco Petitta

Publications and source records attributed to Francesco Petitta.

At least 19 recordsLinked to original sources

arXiv2026

New definitions of a solution of a one-dimensional elliptic equation with a singular first order divergence term

In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as $$ \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d ϕ(u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L),\\ \newline u(0)=u(L)=0\,, & \end{cases} $$ where $ϕ:\mathbb{R}\mapsto\mathbb{R}\cup\{+\infty\}$ is a singular function whose model is given by $\displaystyleϕ(s)=ϕ_γ(s)=\frac{1}{|s|^γ}$ with $γ>0$. In these two definitions the solutions belong respectively to the space $u\in W_0^{1,1}(0,L)$ with $ϕ(u)\in L^1(0,L)$, and to the space $u\in W_0^{1,m}(0,L)$ with $ϕ(u)\in L^m(0,L)$ for $m>1$. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.

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arXiv2025

Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \end{cases} \end{equation} where $Ω$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $Δ_p u$ is the $p$-Laplacian with $p>1$, $0<γ\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $ f\in L^{1}(Ω)$ we show that problem \eqref{eintro} is solvable for any $μ>μ_0>0$, for some $μ_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<μ<μ_{0*}$, for some small $μ_{0*}$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $γ=1$).

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arXiv2025

Quasilinear elliptic equations with singular quadratic growth terms

In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{(1-u)^γ}=g & \mbox{in $Ω$,}\newline \hfill u=0 \hfill & \mbox{on $\partialΩ$,} \end{cases} $$ where $Ω$ is a bounded open set of $\mathbb{R}^N$, $g\geq 0 $ is a function in some Lebesgue space, and $γ>0$. We prove both existence and nonexistence of solutions depending on the value of $γ$ and on the size of $g$.

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arXiv2025

A duality approach to the fractional Laplacian with measure data

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-Δ)^s v = μ\quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $μ$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(Δ)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.

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arXiv2025

Diffuse measures and nonlinear parabolic equations

Given a parabolic cylinder $Q =(0,T)\timesΩ$, where $Ω\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-Δ_p u = μ\quad \text{in $Q$} \] with Dirichlet boundary conditions, where $μ$ is a finite Radon measure in $Q$. We first prove a priori estimates on the $p$-parabolic capacity of level sets of $u$. We then show that diffuse measures (i.e.\@ measures which do not charge sets of zero parabolic $p$-capacity) can be strongly approximated by the measures $μ_k = (T_k(u))_t-Δ_p(T_k(u))$, and we introduce a new notion of renormalized solution based on this property. We finally apply our new approach to prove the existence of solutions of $$ u_t-Δ_{p} u + h(u)=μ\quad \text{in $Q$,} $$ for any function $h$ such that $h(s)s\geq 0$ and for any diffuse measure $μ$; when $h$ is nondecreasing we also prove uniqueness in the renormalized formulation. Extensions are given to the case of more general nonlinear operators in divergence form.

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arXiv2025

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $Ω$. As a consequence of our main results, we prove that the condition $γ<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(Ω)$ for every sufficiently regular $f$ as above.

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arXiv2025

Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - Δ_1 u= h(u)f & \text{in } Ω, \\ \newline u=0 & \text{on } \partial Ω, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(Ω)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(Ω)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator.

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arXiv2025

Global existence for a Leibenson type equation with reaction on Riemannian manifolds

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = Δ_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1 1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.

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arXiv2024

The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities

In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -Δ_1 u = F(x,u) & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} \end{equation*} where $Ω$ is an open bounded domain of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary and $F(x,s)$ is a Caratheódory function. This goal is achieved through a perturbation method that overcomes structural obstructions arising from the presence of the $1$-Laplacian and by proving a weak comparison principle for these problems. As a significant application of our main result we establish existence and non-existence theorems for the so-called ``concave-convex'' problem involving the $1$-Laplacian as leading term.

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arXiv2024

Existence and regularity of solutions for the elliptic nonlinear transparent media equation

In this paper we study existence and regularity of solutions to Dirichlet problems as $$ \begin{cases} - {\rm div}\left(|u|^m\frac{D u}{|D u|}\right) = f & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $m>0$, and $f$ belongs to the Lorentz space $L^{N,\infty}(Ω)$. In particular, we explore the regularizing effect given by the degenerate coefficient $|u|^m$ in order to get non-trivial and bounded solutions with no smallness assumptions on the size of the data.

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arXiv2024

Singular Elliptic PDEs: an extensive overview

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -Δu = {u^{-γ}} \ \text{in}\ Ω$$ where $Ω$ is a bounded subset of $\mathbb{R}^N$ ($N\geq 2$), and $γ>0$. We start by outlining the basic concepts and the mathematical framework needed for setting the problem. Both old and new key existence and uniqueness results are presented, alongside regularity issues depending on the regularity of the data. The presentation aims to be modern, self-contained and consistent. Some examples and open problems are also discussed.

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arXiv2024

Unexpected phenomena in a one dimensional elliptic equation with a singular first order divergence term

We study existence of a weak solution for one-dimensional problems as \begin{equation}\label{intro}\tag{1} \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d ϕ(u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L), u(0)=u(L)=0\,, & \end{cases} \end{equation} where $a$ is a positive bounded function, $g\in L^2(0,L)$, and $ϕ:\mathbb{R}\mapsto \mathbb{R}\cup \{+\infty\}$ is continuous as a function with values in $\mathbb{R}\cup \{+\infty\}$. Some relevant qualitative and quantitative facts concerning such problems and their solutions are described. In particular a precise characterization of the behaviour of suitable approximating solution is provided. Of particular (and independent) interest is the study of an associated ODE for which, we prove existence, uniqueness and comparison results. As a consequence of our arguments, a delicate stability result as well a quite unexpected multiplicity result is shown for problems as in \eqref{intro}.

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arXiv2024

Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $Ω$,} \newline u\geq 0 & \text{in $Ω$,} \newline u=0 & \text{on $\partial Ω$,} \ \end{cases} \end{equation*} in a domain $Ω\subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(Ω)$.

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arXiv2024

Existence, non-existence and degeneracy of limit solutions to $p$-Laplace problems involving Hardy potentials as $p\to1^+$. The case of a critical drift

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_pu&=&λ|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary containing the origin, $λ\in\mathbb{R}$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$. As a consequence, under suitable smallness assumptions on $f$ and $λ$, we show sharp existence results of bounded solutions to the Dirichlet problems $$\begin{cases} \displaystyle - Δ_{1} u = λ\frac{D u}{|D u|}\cdot \frac{x}{|x|^2}+f & \text{in}\, Ω, u=0 & \text{on}\ \partial Ω, \end{cases} $$ where $\displaystyle Δ_{1}u=\hbox{div}\,\left(\frac{Du}{|Du|}\right) $ is the $1$-Laplacian operator. The case of a generic drift term in $L^{N,\infty}(Ω)$ is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.

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arXiv2024

Bounded solutions for non-parametric mean curvature problems with nonlinear terms

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $Ω$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(Ω)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples.

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arXiv2024

Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_p u_p&=&\fracλ{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $λ\in\mathbb{R}^+$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$. Under sharp smallness assumptions on the data $λ$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\left\{ \begin{array}{rclr}- Δ_{1} u &=& \fracλ{|x|}{\rm Sgn}(u)+f & \text{in}\, Ω,\\ u&=&0 & \text{on}\ \partial Ω,\end{array}\right. $$ where $Δ_{1} u ={\rm div}\left(\frac{D u}{|Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.

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arXiv2023

The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} g(u) -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $g:\mathbb{R}\to\mathbb{R}$ is a continuous function and $f$ belongs to some Lebesgue spaces. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term $g(u)$ in order to get a solutions for data $f$ merely belonging to $L^1(Ω)$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^{N}(Ω)$ as well as uniqueness if $g$ is increasing.

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arXiv2021

Nonlinear diffusion in transparent media

We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions' support and in the bulk.

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