arXiv subjects

Hamilton Bueno

Publications and source records attributed to Hamilton Bueno.

14 recordsLinked to original sources

arXiv2025

On compact embeddings in $\mathbf{L^p}$ and fractional spaces

The study of the fractional Laplacian operator $(-Δ)^s$ in $\mathbb{R}^N$ with Dirichlet boundary conditions gained enormous momentum through its identification with a Neumann operator in $\mathbb{R}^N\times (0, \infty)=\mathbb{R}^{N+1}_+$, a method mainly introduced by Caffarelli and Silvestre. Since then, several other operators have been studied using this method. In general, a crucial question is attached to this method: the embedding (in the trace sense) on the ground space $L^q(\mathbb{R}^{N})$ is compact? This question is very important when dealing with problems of existence of solutions. This paper aims to answer this question for some operators. Passing to an abstract setting, let $X,Y$ be Hilbert spaces and $\mathcal{A}\colon X\to X'$ a continuous and symmetric elliptic operator. We suppose that $X$ is dense in $Y$ and that the embedding $X\subset Y$ is compact. In this paper we show some consequences of this setting for the study of the fractional operator attached to $\mathcal{A}$ in the extension setting $Ω\times(0,\infty)$ or $\mathbb{R}^{N+1}_+$. Being more specific, we will give some examples where the embedding of the extension domain into $L^2(Ω)$ is compact, even in the case $Ω=\mathbb{R}^N$.

math.FA↗

arXiv2022

Existence of nonnegative solutions for fractional Schrödinger equations with Neumann condition

In this paper we study a Neumann problem for the fractional Laplacian, namely \begin{equation}\left\{ \begin{array}{rcll} \varepsilon^{2s}(- Δ)^{s}u + u &=& f(u) \ \ &\mbox{in} \ \ Ω\\ \mathcal{N}_{s}u &=& 0 , \,\, &\text{in} \,\, \mathbb{R}^{N}\backslash Ω\end{array}\right. \end{equation} where $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain, $N>2s$, $s \in (0,1)$, $\varepsilon > 0$ is a parameter and $\mathcal{N}_{s}$ is the nonlocal normal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establish the existence of a nonnegative, non-constant small energy solution $u_{\varepsilon}$, and we use the Moser-Nash iteration procedure to show that $u_{\varepsilon} \in L^{\infty}(Ω)$.

math.AP↗

arXiv2021

Multiplicity of solutions for a scalar field equation involving a fractional $p$-Laplacian with general nonlinearity

We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Δ_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-Δ_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.

math.AP↗

arXiv2020

Critical concave convex Ambrosetti-Prodi type problem for fractional $p$-Laplacian

In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the nonlinearities with the first eigenvalue of fractional $p$-Laplacian will be used to prove existence and multiplicity of solutions.

math.AP↗

arXiv2020

Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition

In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a Ambrosetti-Rabinowitz-type condition. The interaction of the nonlinearities with the spectrum of the fractional operator will used to study the existence and multiplicity of solutions. The main technical result proves that a local minimum in $C_{s}^0(\overlineΩ)$ is also a local minimum in $W^{s,p}_0$ for nonlinearities with exponential growth.

math.AP↗

arXiv2020

Torsion functions and the Cheeger problem: a fractional approach

Let $Ω$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The fractional Cheeger constant $h_s (Ω)$, $0<s<1$, is defined by \[h_s(Ω)=\inf_{E\subsetΩ}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N }\frac{|χ_{E}(x)-χ_{E}(y)|}{|x-y|^{N+s}} dx dy,\] with $χ_{E}$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that \[\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},\] where $ϕ_p^s$ is the fractional $(s,p)$-torsion function of $Ω$, that is, the solution of the Dirichlet problem for the fractional $p$-Laplacian: $-(Δ)_p^s\,u=1$ in $Ω$, $u=0$ in $\mathbb{R}^N \setminusΩ$. For this, we derive suitable bounds for the first eigenvalue $λ_{1,p}^s(Ω)$ of the fractional $p$-Laplacian operator in terms of $ϕ_p^s$. We also show that $ϕ_p^s$ minimizes the $(s,p)$-Gagliardo seminorm in $\mathbb{R}^N $, among the functions normalized by the $L^1$-norm.

math.AP↗

arXiv2019

Nonlinear Perturbations of a periodic magnetic Choquard equation with Hardy-Littlewood-Sobolev critical exponent

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^α}*|u|^{2_α^*}\right) |u|^{2_α^*-2} u + λf(u)\ \textrm{ in }\ \R^N,\] where $2_α^{*}=\frac{2N-α}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, $λ>0$, $N\geq 3$, $0<α< N$, $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is an $C^1$, $\mathbb{Z}^N$-periodic vector potential and $V$ is a continuous scalar potential given as a perturbation of a periodic potential. Under suitable assumptions on different types of nonlinearities $f$, namely, $f(x,u)=\left(\frac{1}{|x|^α}*|u|^{p}\right)|u|^{p-2} u$ for $(2N-α)/N<p<2^{*}_α$, then $f(u)=|u|^{p-1} u$ for $1<p<2^*-1$ and $f(u)=|u|^{2^* - 2}u$ (where $2^*=2N/(N-2)$), we prove the existence of at least one ground state solution for this equation by variational methods if $p$ belongs to some intervals depending on $N$ and $λ$.

math.AP↗

arXiv2019

Existence, regularity, asymptotic decay and radiality of solutions to some extension problems

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem \begin{equation*}\left\{ \begin{array}{rcll} -Δu+ m^2u &=& 0, &\mbox{in} \ \ \mathbb{R}^{N+1}_{+} \\ -\frac{\partial u}{\partial{x}} (0,y)& =& f(u(0,y)), & y \in \mathbb{R}^{N}, \end{array}\right. \end{equation*} and also to the extension Hartree problem \begin{equation*} \left\{\begin{aligned} -Δu +m^2u&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial u}{\partial x}(0,y)&=-V_\infty u(0,y)+\left(\frac{1}{|y|^{N-α}}*F(u(0,y))\right)f(u(0,y)) &&\mbox{in} \ \mathbb{R}^{N}\end{aligned}\right. \end{equation*} are radially symmetric in $\mathbb{R}^N$. In the last problem, $V_\infty>0$ is a constant and $F$ the primitive of $f$. Under the same hypotheses, regularity and exponential decay of solutions to the first problem is also proved and, supposing the traditional Ambrosetti-Rabinowitz condition, also existence of a ground state solution.

math.AP↗

arXiv2018

Remarks about a generalized pseudo-relativistic Hartree equation

With appropriate hypotheses on the nonlinearity $f$, we prove the existence of a ground state solution $u$ for the problem \[(-Δ+m^2)^σu+Vu=\left(W*F(u)\right)f(u)\ \ \text{in }\ \mathbb{R}^{N},\] where $0<σ<1$, $V$ is a bounded continuous potential and $F$ the primitive of $f$. We also show results about the regularity of any solution of this problem.

math.AP↗

arXiv2018

Ground state of a magnetic nonlinear Choquard equation

We consider the stationary magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u=\bigg(\frac{1}{|x|^α}*F(|u|)\bigg)\frac{f(|u|)}{|u|}{u},\] where $A: \mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a vector potential, $V$ is a scalar potential, $f\colon\mathbb{R}\to\mathbb{R}$ and $F$ is the primitive of $f$. Under mild hypotheses, we prove the existence of a ground state solution for this problem. We also prove a simple multiplicity result by applying Ljusternik-Schnirelmann methods.

math.AP↗

arXiv2012

A quasilinear problem with fast growing gradient

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $λ$ and $β$: [{{array} [c]{rcll}% -Δ_{p}u & = & λh(x,u)+βf(x,u,\nabla u) & \text{in}Ωu & = & 0 & \text{on}\partialΩ, {array}. \hfill] where $h,f$ are continuous nonlinearities satisfying $0\leqω_{1}(x)u^{q-1}\leq h(x,u)\leqω_{2}(x)u^{q-1}$ with $1 0$, and $Ω$ is a bounded domain of $\mathbb{R}^{N},$ $N\geq3.$ The functions $ω_{i}$, $1\leq i\leq3$, are nonnegative, continuous weights in $\barΩ$. We prove that there exists a region $\mathcal{D}$ in the $λβ$-plane where the Dirichlet problem has at least one positive solution. The novelty in this paper is that our result is valid for nonlinearities with growth higher than $p$ in the gradient variable.

math.AP↗

arXiv2011

Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain $Ω\subset\mathbb{R}^{N}$, $N>1$ consists in minimizing the quotients $|\partial E|/|E|$ among all smooth subdomains $E\subsetΩ$ and the Cheeger constant $h(Ω)$ is the minimum of these quotients. Let $ϕ_{p}\in C^{1,α}(\barΩ)$ be the $p$-torsion function, that is, the solution of torsional creep problem $-Δ_{p}ϕ_{p}=1$ in $Ω$, $ϕ_{p}=0$ on $\partialΩ$, where $Δ_{p}u:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator, $p>1$. The paper emphasizes the connection between these problems. We prove that $\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{\infty}(Ω)})^{1-p}=h(Ω)=\lim_{p\rightarrow1^{+}}(\|ϕ_{p}\|_{L^{1}(Ω)})^{1-p}$. Moreover, we deduce the relation $\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{1}(Ω)}\geq C_{N}\lim_{p\to1^{+}}\|ϕ_{p}\|_{L^{\infty}(Ω)}$ where $C_{N}$ is a constant depending only of $N$ and $h(Ω)$, explicitely given in the paper. An eigenfunction $u\in BV(Ω)\cap L^{\infty}(Ω)$ of the Dirichlet 1-Laplacian is obtained as the strong $L^{1}$ limit, as $p\rightarrow1^{+}$, of a subsequence of the family $\{ϕ_{p}/\|ϕ_{p}\|_{L^{1}(Ω)}\}_{p>1}$. Almost all $t$-level sets $E_{t}$ of $u$ are Cheeger sets and our estimates of $u$ on the Cheeger set $|E_{0}|$ yield $|B_{1}|h(B_{1})^{N}\leq |E_{0}|h(Ω)^{N},$ where $B_{1}$ is the unit ball in $\mathbb{R}^{N}$. For $Ω$ convex we obtain $u=|E_{0}|^{-1}χ_{E_{0}}$.

math.AP↗

arXiv2010

Positive Solutions for the p-Laplacian with Dependence on the Gradient

We prove a result of existence of positive solutions of the Dirichlet problem for $-Δ_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $Ω\subset\mathbb{R}^N$, where $Δ_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on $f$, but simple geometric assumptions on a neighborhood of the first eigenvalue of the $p$-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub- and super-solution, also valid for nonlinearities which are super-linear both at the origin and at $+\infty$. We apply our method to the Dirichlet problem $-Δ_pu = λu(x)^{q-1}(1+|\nabla u(x)|^p)$ in $Ω$ and give examples of super-linear nonlinearities which are also handled by our method.

math.AP↗

arXiv2010

A quasilinear problem in two parameters depending on the gradient

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -Δ_{p}u & = & λω_{1}(x)\left\vert u\right\vert ^{q-2}% u+βω_{2}(x)\left\vert u\right\vert ^{a-1}u|\nabla u|^{b} & \text{in }Ω\\ u & = & 0 & \text{on }\partialΩ, \end{array} \right. \] where $λ$ and $β$ are positive parameters, $a$ and $b$ are positive constants satisfying $a+b\leq p-1$, $ω_{1}(x)$ and $ω_{2}(x)$ are nonnegative weights and $1<q\leq p$. The homogeneous case $q=p$ is handled by making $q\rightarrow p^{-}$ in the sublinear case $1<q<p,$ which is based on the sub- and super-solution method. The core of the proof of this problem is then generalized to the Dirichlet problem $-Δ_{p}u=f(x,u,\nabla u)$ in $Ω$, where $f$ is a nonnegative, continuous function satisfying simple, geometrical hypotheses. This approach might be considered as a unification of arguments dispersed in various papers, with the advantage of handling also nonlinearities that depend on the gradient, even in the $p$-growth case. It is then applied to the problem $-Δ_{p}u=λω(x)u^{q-1}\left( 1+|\nabla u|^{p}\right) $ with Dirichlet boundary conditions in the domain $Ω$.

math.AP↗