arXiv subjects

Alexis Molino

Publications and source records attributed to Alexis Molino.

8 recordsLinked to original sources

arXiv2026

Lower order term for the fractional Laplacian with a Hardy potential

This paper is concerned with the following semilinear elliptic equation involving the fractional Laplacian: $$(-\Delta)^s u+ g|u|^{p-1}u= \lambda \frac{u}{|x|^{2s}}+f(x),$$ in a bounded domain $\Omega$ of $\mathbb{R}^N\,(N>2s)$, subject to the zero Dirichlet condition in $\mathbb{R}^N\setminus \Omega$, where $0 1$ and $f\in L^{(p+1)/p}_g(\Omega)$. Under certain integrability condition on $g$, the existence of solution is proven for every $\lambda\in \mathbb{R}$. Moreover, the regularity of solution is also obtained.

math.AP↗

arXiv2025

Solutions for autonomous semilinear elliptic equations

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $\lambda \in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ is any locally Lipschitz function with nonpositive primitive. A complete description is obtained for $N=1$ and partial results for $N\geq 2$.

math.AP↗

arXiv2020

Gelfand type problems involving the 1-Laplacian operator

In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -Δ_1u=λf(u) &\hbox{in }Ω\,;\\[2mm] u=0 &\hbox{on }\partialΩ\,; \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $λ\geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. It is proved the existence of a threshold $λ^*=\frac{h(Ω)}{f(0)}$ (being $h(Ω)$ the Cheeger constant of $Ω$) such that there exists no solution when $λ>λ^*$ and the trivial function is always a solution when $λ\leλ^*$. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour of solutions to problems involving the $p$--Laplacian as $p$ tends to 1, which allows us to identify proper solutions through an extra condition.

math.AP↗

arXiv2019

Nonexistence result for a semilinear elliptic problem

In this paper we prove the nonexistence of nontrivial solution to \begin{equation*} \begin{cases} -Δu =f(u) &\text{in }Ω, \\ u=0 &\text{on } \partial Ω, \end{cases} \end{equation*} being $Ω\subset \mathbb{R}^N$ ($N\in \mathbb{N}$) a bounded domain and $f$ locally Lispchitz with non-positive primitive.

math.AP↗

arXiv2017

Elliptic equations involving the $1$-Laplacian and a subcritical source term

In this paper we deal with a Dirichlet problem for an elliptic equation involving the $1$-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohozaev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results.

math.AP↗

arXiv2017

A concave-convex problem with a variable operator

We study the following elliptic problem $-A(u) = λu^q$ with Dirichlet boundary conditions, where $A(u) (x) = Δu (x) χ_{D_1} (x)+ Δ_p u(x) χ_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2 $. We show that this problem exhibits a concave-convex nature for $1 λ^*$ and a minimal positive solution for $0<λ< λ^*$. If in addition we assume that $p$ is subcritical, that is, $p<2N/(N-2)$ then there are at least two positive solutions for almost every $0<λ< λ^*$, the first one (that exists for all $0<λ< λ^*$) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $0<λ< λ^*$) comes from an appropriate (and delicate) mountain pass argument.

math.AP↗

arXiv2017

Parabolic equations with natural growth approximated by nonlocal equations

In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y) \big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad \mbox{ in } \, Ω\times (0,T)\,, $$ being $ u (x,t)=0 \mbox{ in } (\mathbb{R}^N\setminus Ω)\times (0,T)\,$ and $ u(x,0)=u_0 (x) \mbox{ in } Ω$. We take, as the most important instance, $\mathcal G (s) \sim 1+ \fracμ{2} \frac{s}{1+μ^2 s^2 }$ with $μ\in \mathbb{R}$ as well as $u_0 \in L^1 (Ω)$, $J$ is a smooth symmetric function with compact support and $Ω$ is either a bounded smooth subset of $\mathbb{R}^N$, with nonlocal Dirichlet boundary condition, or $\mathbb{R}^N$ itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover we prove that if the kernel rescales in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar-Parisi-Zhang equation.

math.AP↗

arXiv2013

Local triple derivations on real C-algebras and JB-triples

We study when a local triple derivation on a real JB*-triple is a triple derivation. We find an example of a (real linear) local triple derivation on a rank-one Cartan factor of type I which is not a triple derivation. On the other hand, we find sufficient conditions on a real JB*-triple E to guarantee that every local triple derivation on E is a triple derivation.

math.OA↗