arXiv subjects

César E. Torres Ledesma

Publications and source records attributed to César E. Torres Ledesma.

7 recordsLinked to original sources

arXiv2023

Fractional tempered variational calculus

In this paper, we derive sufficient conditions ensuring the existence of a weak solution $u$ for a tempered fractional Euler-Lagrange equations $$ \frac{\partial L}{\partial x}(u,{^C}\mathbb{D}_{a^+}^{α, σ} u, t) + \mathbb{D}_{b^-}^{α, σ}\left(\frac{\partial L}{\partial y}(u, {^C}\mathbb{D}_{a^+}^{α, σ}u, t) \right) = 0 $$ on a real interval $[a,b]$ and ${^C}\mathbb{D}_{a^+}^{α, σ}, \mathbb{D}_{b^-}^{α, σ}$ are the left and right Caputo and Riemann-Liouville tempered fractional derivatives respectively of order $α$. Furthermore, we study a fractional tempered version of Noether theorem and we provide a very explicit expression of a constant of motion in terms of symmetry group and Lagrangian for fractional problems of calculus of variations. Finally we study a mountain pass type solution of the cited problem.

math.AP↗

arXiv2020

Existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ with critical growth: The local Mountain pass case

In this paper we study the existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ where the nonlinearity has a critical growth for $N \geq 2$. By combining a penalization scheme found in del Pino and Felmer [18] with a penalization method due to Bensoussan and Lions [9], we improve a recent result by Alves, Barros and Torres [1].

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arXiv2019

Multiplicity of solutions for a class of fractional elliptic problem with critical exponential growth and nonlocal Neumann condition

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R \setminus (a,b)\\ \mathcal{N}_{1/2}u(x) &= 0\;\;\mbox{in}\;\;(a,b), \end{aligned} \right. \end{equation} where $a,b\in \R$ with $a<b$, $(-Δ)^{\frac{1}{2}}$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_{1/2}u(x) = \frac{1}π \int_{\R\setminus (a,b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy,\;\;x\in [a,b]. $$

math.AP↗

arXiv2016

Existence and concentration of solution for a non-local regional Schrödinger equation with competing potentials

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$ \left\{ \begin{array}{l} ε^{2α}(-Δ)_ρ^α u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^α(\mathbb{R}^n) \end{array} \right. $$ where $ε$ is a positive parameter, $0< α< 1$, $1 2α$; $(-Δ)_ρ^α$ is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius $ρ(x)>0$, $ρ, Q, K$ are competing functions. We study the existence of ground state and we analyze the behavior of semi-classical solutions as $ε\to 0$.

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arXiv2016

Existence and multiplicity of solutions for a nonlinear Schrödinger equation with non-local regional diffusion

In this article we are interested in the following non-linear Schrödinger equation with non-local regional diffusion $$ (-Δ)_{ρ_ε}^αu + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^α(\mathbb{R}^n), \qquad\qquad(P_ε) $$ where $ε>0$, $0< α< 1$, $(-Δ)_{ρ_ε}^α$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $ρ_ε(x)=ρ(εx)>0$, where $ρ$ is a continuous function. We give general conditions on $ρ$ and $f$ which assure the existence and multiplicity of solution for $(P_ε)$.

math.AP↗

arXiv2016

Concentration of ground state solution for a fractional Hamiltonian Systems

In this paper we are concerned with the existence of ground states solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} -_tD^α_\infty(_{-\infty}D^α_t u(t)) - λL(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u \in H^α(\mathbb{R},\mathbb{R}^n), \end{array} \right.\qquad(\hbox{FHS})_λ$$ where $α\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $λ>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that (FHS)$_λ$ has a ground sate solution which vanishes on $\mathbb{R}\setminus T$ as $λ\to \infty$, and converges to $u\in H^α(\mathbb{R}, \mathbb{R}^n)$, where $u\in E_{0}^α$ is a ground state solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Recent results are generalized and significantly improved.

math.AP↗

arXiv2015

Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

This article study the fractional Hamiltonian systems \begin{eqnarray}\label{00} {_{t}}D_{\infty}^α({_{-\infty}}D_{t}^αu) + λL(t)u = \nabla W(t, u), \;\;t\in \mathbb{R}, \end{eqnarray} where $α\in (1/2, 1)$, $λ>0$ is a parameter, $L\in C(\mathbb{R}, \mathbb{R}^{n\times n})$ and $W \in C^{1}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R})$. Unlike most other papers on this problem, we require that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t) \equiv 0$ is allowed to occur in some finite interval $\mathbb{I}$ of $\mathbb{R}$. Under some mild assumptions on $W$, we establish the existence of nontrivial weak solution, which vanish on $\mathbb{R} \setminus \mathbb{I}$ as $λ\to \infty,$ and converge to $\tilde{u}$ in $H^α(\mathbb{R})$; here $\tilde{u} \in E_{0}^α$ is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval $\mathbb{I}$.

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