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César Torres

Publications and source records attributed to César Torres.

11 recordsLinked to original sources
arXiv2018

Existence and concentration of solution for a fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^α_{\infty}(_{-\infty}D^α_{t}u(t))-λL(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^α(\mathbb{R},\mathbb{R}^n), \end{array} \right. \eqno(\mbox{FHS})_λ$$ 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 $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. 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 some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_λ$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $λ\to \infty$, and converges to some $\tilde{u}\in H^α(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^α$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $α=1$.

math.AP↗
arXiv2017

A critical nonlinear elliptic equation with non local regional diffusion

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent \begin{eqnarray*} &ε^{2α} (-Δ)_ρ^αu + u = λu^q + u^{2_α^{*}-1} \mbox{ in } \mathbb{R}^{N}, \\ & u \in H^α(\mathbb{R}^{N}), \end{eqnarray*} where $ε$ is a small positive parameter, $α\in (0,1)$, $q\in (1,2_α^{*}-1)$, $2_α^{*} = \frac{2N}{N-2α}$ is the critical Sobolev exponent, $λ>0$ is a parameter and $(-Δ)_ρ^α$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $ρ(x)>0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon\to 0$.

math.AP↗
arXiv2017

Existence and multiplicity result for a fractional p-Laplacian equation with combined fractional derivatives

The aim of this paper is to obtain the existence of solutions for the following fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*} &{_{t}}D_{T}^α\left(|_{0}D_{t}^αu(t)|^{p-2}{_{0}}D_{t}^αu(t)\right) = f(t, u(t)), \;t\in [0,T],\\ &u(0) = u(T) = 0, \end{eqnarray*} where $0 < α<1$, $1<p<\infty$ and $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a continuous function. We obtain the existence of nontrivial solutions by using the direct method in variational methods and the genus in the critical point theory. Furthermore, if $0< α< \frac{1}{p}$ we obtain an almost every where classical solution.

math.AP↗
arXiv2016

Existence and symmetry result for Fractional p-Laplacian in $\mathbb{R}^{n}$

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-Δ)_{p}^αu + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s < 1$, $n\geq 2$ and subcritical p-superlinear nonlinearity. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.

math.AP↗
arXiv2015

Non-homogeneous fractional Schrödinger equation

In this article we are interested on the non-homogeneous fractional Schrödinger equation \begin{eqnarray}\label{eq00} &(-Δ)^αu(x) + V(x)u(x) = f(u) + h(x) \mbox{ in } \mathbb{R}^{n}. \end{eqnarray} By using mountain pass Thoerem and Ekeland's variational principle, we prove the existence of two nontrivial solutions for (\ref{eq00}).

math.AP↗
arXiv2014

Boundary value problem with fractional p-Laplacian operator

The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives \begin{eqnarray*} &{_{t}}D_{T}^α\left(|_{0}D_{t}^αu(t))|^{p-2}{_{0}}D_{t}^αu(t)\right) = f(t,u(t)), \;t\in [0,T],\\ &u(0) = u(T) = 0, \end{eqnarray*} where $\frac{1}{p} < α<1$, $1<p<\infty$ and $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function wich satisfies some growth conditions. We obtain the existence of nontrivial solution by using the Mountain Pass Theorem.

math.AP↗
arXiv2014

Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation

We study the existence of positive solution for the one dimensional Schrödinger equation with mixed Lioville-Weyl fractional derivatives \begin{eqnarray*}\label{Eq00} _{t}D_{\infty}^α({_{-\infty}}D_{t}^αu(t)) + V(t) u(t) = & f(u(t)),\;\;t\in \mathbb{R}\\ u\in H^α(\mathbb{R}).\nonumber \end{eqnarray*} Furthermore, we analyse radial symmetry property of these solutions. The proof is carried out by using variational methods jointly with comparison and rearrangement argument.

math-ph↗
arXiv2014

Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivative

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00} _{t}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) + L(t)u(t) = & \nabla W(t,u(t))\\ u\in H^α(\mathbb{R}, \mathbb{R}^{N}).\nonumber \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^2})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$, and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming there exists $l\in C(\mathbb{R}, \mathbb{R})$ such that $(L(t)u,u)\geq l(t)|u|^{2}$ for all $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$ and the following conditions on $l$: $\inf_{t\in \mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that, for any $M>0$ $$ m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) \to 0\;\;\mbox{as}\;\;|y|\to \infty. $$ are satisfied and $W$ is of subquadratic growth as $|u| \to +\infty$, we show that (\ref{eq00}) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in [Z. Zhang and R. Yuan, Solutions for subquadratic fractional Hamiltonian systems without coercive conditions, Math. Methods Appl. Sci., DOI: 10.1002/mma.3031] are significantly improved.

math-ph↗
arXiv2014

Existence of solution for perturbed fractional Hamiltonian systems

In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^α(_{-\infty}D_{t}^αu(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t), \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$ and $\nabla W$ is the gradient of $W$ at $u$. The novelty of this paper is that, assuming $L$ is coercive at infinity we show that (\ref{eq00}) at least has one nontrivial solution.

math.AP↗
arXiv2013

Ground state solution for a class of differential equations with left and right fractional derivatives

In this work we study the existence of solution for a class of fractional differential equation given by \begin{eqnarray}\label{eq00} _{t}D_{\infty}^α{_{-\infty}}D_{t}^αu(t) + u(t) = & f(t,u(t))\\ u\in H^α(\mathbb{R}).\nonumber \end{eqnarray} where $α\in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}$, $f\in C(\mathbb{R}, \mathbb{R})$. Using mountain pass theorem and comparison argument we prove that (\ref{eq00}) at least has one nontrivial solution.

math.AP↗