arXiv subjects

Cesar E. Torres Ledesma

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

2 recordsLinked to original sources

arXiv2019

Fractional elliptic problem in exterior domains with nonlocal Neumann boundary condition

In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is a bounded set with smooth boundary, $(-Δ)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus Ω} \frac{u(x) - u(y)}{|x-y|^{N+2s}}dy,\;\;x\in Ω. $$

math.AP↗

arXiv2018

Existence of solutions for a class of fractional elliptic problems on exterior domains

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-Δ)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $Ω$},\\ u \geq 0 \quad \mbox{in} \quad Ω\quad \mbox{and} \quad u \not\equiv 0, \\ u=0 \quad \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation*} involving the fractional Laplacian operator $(-Δ)^{s}$, where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is an exterior domain with (non-empty) smooth boundary $\partial Ω$ and $p\in (2, 2_{s}^{*})$. The main technical approach is based on variational and topological methods. The variational analysis that we perform in this paper dealing with exterior domains is quite general and may be suitable for other goals too.

math.AP↗