arXiv subjects

Antonio J. Martínez Aparicio

Publications and source records attributed to Antonio J. Martínez Aparicio.

8 recordsLinked to original sources

arXiv2026

Existence and classification of solutions for singular 1-Laplacian equations with gradient terms

We study the existence and qualitative properties of non-negative solutions to singular 1-Laplacian problems of the form \begin{equation*} \begin{cases} -Δ_1 u = g(u)|D u| + h(u)f & \text{in } Ω,\newline u=0 & \text{on } \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<f\in L^{N,\infty}(Ω)$, and $g,h$ are continuous positive functions that may blow up at the origin. We establish the existence of a positive bounded solution even when $g$ is non-integrable at zero and $h(0)=+\infty$, breaking the non-existence phenomenon appearing in the associated $p$-Laplacian problems. Furthermore, we prove that solutions possess a non-zero boundary trace on a set of positive measure of $\partialΩ$ whenever $g(s)$ grows sufficiently fast near $s=0$; by contrast, when this growth condition fails, solutions vanishing on the boundary may appear. Finally, when $h(0)<+\infty$, we provide a sharp threshold classification of trivial and non-trivial solutions depending on the size of $f$.

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arXiv2026

Existence and comparison results for a doubly singular 1-Laplacian problem with $L^1$ data

In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -Δ_1 u + g(u)|Du| = h(u)f & \text{in }Ω, u=0 & \text{on } \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(Ω)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(Ω)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.

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arXiv2026

Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -Δ_p u + g(u) |\nabla u|^p &= λf(u) \quad &&\mbox{in} \;\; Ω, \\ u &= 0 \quad &&\mbox{on} \;\; \partialΩ, \end{alignedat} \right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $p\in (1,\infty)$, $λ>0$ and $g\in C(\mathbb{R})$. Our main assumption is that $:f \mathbb{R}\to \mathbb{R}$ is a continuous function such that $f(s)>0$ for all $s\in (α,β)$, where $0<α<β$ are two zeros of $f$. If $f(0)\geq 0$, we show that an area condition involving $f$ and $g$ is both sufficient and necessary in order to have a pair $(λ,u)\in \mathbb{R}^+\times C_0^1(\overlineΩ)$, with $u\geq 0$ and $\|u\|_{C(\overlineΩ)}\in (α,β]$, solving~\eqref{pba}. We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative $g$ is, the stronger its regularizing effect on~\eqref{pba}. We prove that, regardless of the shape of $f$, for any fixed $λ$, there always exists a function $g$ such that~\eqref{pba} admits a nonnegative solution with maximum in $(α,β]$.

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arXiv2026

Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-Δ_p u = λf(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function with a sequence of positive zeros converging to zero or diverging to infinity. Under a growth condition on the primitive $F(s) = \int_0^s f(t)dt$, we establish ranges of the parameter $λ$ for which infinitely many small or large solutions exist, as well as ranges where no bifurcation from zero or infinity can occur. The existence result is obtained via variational methods for a general class of divergence form operators, while the nonexistence result is established both for the $p$-Laplacian and for uniformly elliptic operators in non-divergence form via an ODE argument.

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arXiv2025

Multiplicity of solutions for semilinear Robin problems involving sign-changing nonlinearities

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -Δu = λf(u) & \text{in } Ω, \frac{\partial u}{\partial ν} + γu=0 & \text{on } \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N\geq 1$) is a smooth bounded domain, and $λ, γ>0$. Our main assumption is that $f\colon \mathbb{R}\to \mathbb{R}$ is a locally Lipschitz function, possibly sign-changing, such that $f(s)>0$ for every $s\in (α,β)$, where $0<α<β$ are two zeros of $f$. Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in $(α,β)$ for sufficiently large $λ$. Moreover, we analyse the limiting behaviour of the solution set of this Robin problem, showing that it degenerates into that of the associated Neumann problem as $γ\to 0$ and into that of the associated Dirichlet problem as $γ\to\infty$.

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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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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

Intervals of bifurcation points for semilinear elliptic problems

In this paper, we study the behavior of multiple continua of solutions to the semilinear elliptic problem \begin{equation*} \begin{cases} -Δu = λf(u) &\text{ in } Ω, u=0 &\text{ on } \partialΩ, \end{cases} \end{equation*} where $Ω$ is a bounded open subset of $\re^N$ and $f$ is a nonnegative continuous real function with multiple zeros. We analyze both the behavior of unbounded continua of solutions having norm between consecutive zeros of $f$, and the asymptotic behavior of the multiple unbounded continua in the case in which $f$ has a countable infinite set of positive zeros. In both cases, we pay special attention to the multiplicity results they give rise to. For the model cases $f(t) = t^r(1+\sin t)$ and $f(t) = t^r \left(1+\sin \frac{1}{t}\right)$ with $r\geq 0$ we show the surprising fact that there are some values of $r$ for which every $λ>0$ is a bifurcation point (either from infinity or from zero) that is not a branching point.

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