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Francescantonio Oliva

Publications and source records attributed to Francescantonio Oliva.

28 records · Page 2Linked to original sources
arXiv2020

The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -Δ_1 u =h(u)f(x)&\hbox{in }Ω\,,\\ u=0&\hbox{on }\partialΩ\,, \end{array}\right. $$ where $ Ω\subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(Ω)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.

math.AP↗
arXiv2019

Comparison principle for elliptic equations with mixed singular nonlinearities

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -Δ_p u= \frac{f}{u^γ} + g u^q & \mbox{in $Ω$,} \\ u = 0 & \mbox{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $Δ_p u:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-Laplacian operator, $γ\geq 0$ and $0\leq q\leq p-1$; $f$ and $g$ are nonnegative functions belonging to suitable Lebesgue spaces.

math.AP↗
arXiv2019

1-Laplacian type problems with strongly singular nonlinearities and gradient terms

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -Δ_1 u = g(u)|D u|+h(u)f & \text{in}\;Ω,\\ u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(Ω)$, and $g$ and $h$ are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities $g$ and $h$ produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.

math.AP↗
arXiv2019

A nonlinear parabolic problem with singular terms and nonregular data

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$ \begin{cases} \displaystyle u_t - Δ_p u = h(u)f+μ& \text{in}\ Ω\times (0,T),\\ u=0 &\text{on}\ \partialΩ\times (0,T),\\ u=u_0 &\text{in}\ Ω\times \{0\}, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ is a nonnegative integrable function, $Δ_p$ is the $p$-laplace operator, $μ$ is a nonnegative bounded Radon measure on $Ω\times (0,T)$ and $f$ is a nonnegative function of $L^1(Ω\times (0,T))$. The term $h$ is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing $h$.

math.AP↗
arXiv2018

Regularizing effect of absorption terms in singular problems

We prove existence of solutions to problems whose model is $$\begin{cases} \displaystyle -Δ_p u + u^q = \frac{f}{u^γ} & \text{in}\ Ω, \newline u\ge0 &\text{in}\ Ω,\newline u=0 &\text{on}\ \partialΩ, \end{cases}$$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $Δ_p u$ is the $p$-laplacian operator for $1\le p 0$, $γ\ge 0$ and $f$ is a nonnegative function in $L^m(Ω)$ for some $m\ge1$. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case $γ\le 1$. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local $W^{1,p}$-solutions in case $γ>1$.

math.AP↗
arXiv2018

The Dirichlet problem for singular elliptic equations with general nonlinearities

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - Δ_{1} u = h(u)f & \text{in}\, Ω,\newline u\geq 0& \text{in}\ Ω, \newline u=0 & \text{on}\ \partial Ω, \end{cases} $$ where, $Δ_{1} $ is the $1$-laplace operator, $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.

math.AP↗
arXiv2017

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{cases} -Δ_p u = H(u)μ& \text{in}\ Ω,\\ u>0 &\text{in}\ Ω,\\ u=0 &\text{on}\ \partialΩ. \end{cases}$$ Here $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $Δ_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ ($1<p<N$) is the $p$-laplacian operator, $μ$ is a nonnegative bounded Radon measure on $Ω$ and $H(s)$ is a continuous, positive and finite function outside the origin which grows at most as $s^{-γ}$, with $γ\ge0$, near zero.

math.AP↗
arXiv2017

On singular elliptic equations with measure sources

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -Δu = \frac{f(x)}{u^γ} +μ& \text{in}\ Ω, u=0 &\text{on}\ \partialΩ, u>0 &\text{on}\ Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$. Here $γ> 0$, $f$ is a nonnegative function on $Ω$, and $μ$ is a nonnegative bounded Radon measure on $Ω$.

math.AP↗
arXiv2016

Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -Δu= h(u){f} \ \ \text{in}\,\ Ω, $$ where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

math.AP↗