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Lucio Boccardo

Publications and source records attributed to Lucio Boccardo.

12 recordsLinked to original sources
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 $g$.

math.AP↗
arXiv2025

Mosco-convergence of convex sets and unilateral problems for differential operators with lower order terms having natural growth

We study the stability of solutions to a class of variational inequalities posed on obstacle-type convex sets, under Mosco-convergence. More precisely, for a fixed obstacle $ψ\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$, we consider $u\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ satisfying $u\geqψ$ a.e. and $$ \langle A(u),v-u\rangle+\int_ΩH(x,u,\nabla u)(v-u)\geq 0$$ for all $v\in W_{0}^{1,p}(Ω)\cap L^{\infty}(Ω)$ with $v\geqψ$. Here, $A$ is a Leray-Lions type operator, mapping $W_0^{1,p}(Ω)$ into its dual $W^{-1, p'}(Ω)$, while $H(x, u, D u)$ grows like $|D u|^p$. Our main result establishes that the solutions are stable under Mosco-convergence of the constraint sets. This extends classical stability results to natural growth problems.

math.AP↗
arXiv2023

Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly interested in singular $E$ with $\mathrm{div} E \ge 0$. We start by recalling known existence results when $|E| \in L^N$ that do not rely on the sign of $\mathrm{div} E $. Then, under the assumption that $\mathrm{div} E \ge 0$ distributionally, we extend the existence theory to $|E| \in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax /|x|^2$, or towards the boundary as $\mathrm{div} E \sim \mathrm{dist}(x, \partial Ω)^{-2-α}$. In these cases the singularity of $E$ leads to $u$ vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. $\partial u / \partial n < 0$, fails in the presence of such singular drift terms $E$.

math.AP↗
arXiv2021

A singular Schrödinger-Maxwell system

In this paper we are concerned with existence of positive solutions for a Schrödinger-Maxwell system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and controlling the approximated sequences of solutions with suitable barriers from above and from below. Besides this, in some particular case, we show that the unique energy solution of the singular system is a saddle point of a suitable functional.

math.AP↗
arXiv2021

Asymptotic behavior of positive solutions of semilinear elliptic problems with increasing powers

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=λu^{p-1} & \text{ in $Ω$}, \\ \quad u>0 &\text{ in $Ω$}, \\ \quad u=0 & \text{ on $\partial Ω$}, \end{cases} \] where $L(v)=-{\rm div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $λ$ and $ m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviors.

math.AP↗
arXiv2016

The role of interplay between coefficients in the $G$-convergence of some elliptic equations

We study the behavior of the solutions $u$ of the linear Dirichlet problems $- \mathrm{div} (M(x) \nabla u) + a(x) u = f(x)$ with respect to perturbations of the matrix $M(x)$ (with respect to the $G$-convergence) and with respect to perturbations of the nonnegative coefficient $a(x)$ and of the right hand side $f(x)$ satisfying the condition $|f (x)| \leq Q \, a (x)$.

math.AP↗
arXiv2011

A nonlinear degenerate elliptic problem with W^{1,1}_0 solutions

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.

math.AP↗