arXiv subjects

J. García-Melián

Publications and source records attributed to J. García-Melián.

3 recordsLinked to original sources

arXiv2018

Periodic solutions for the one-dimensional fractional Laplacian

In this paper we are concerned with the construction of periodic solutions of the nonlocal problem $(-Δ)^s u= f(u)$ in $\mathbb{R}$, where $(-Δ)^s$ stands for the $s$-Laplacian, $s\in (0,1)$. We introduce a suitable framework which allows to reduce the search for such solutions to the resolution of a boundary value problem in a suitable Hilbert space, thereby making it possible to reach for the usual tools of nonlinear analysis, like bifurcation theory or variational methods. We obtain some existence theorems which are lately enlightened with the analysis of some examples.

math.AP↗

arXiv2017

Symmetry results in the half space for a semi-linear fractional Laplace equation through a one-dimensional analysis

In this paper we analyze the semi-linear fractional Laplace equation $$(-Δ)^s u = f(u) \quad\text{ in } \mathbb{R}^N_+,\quad u=0 \quad\text{ in } \mathbb{R}^N\setminus \mathbb{R}^N_+,$$ where $\mathbb{R}^N_+=\{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ stands for the half-space and $f$ is a locally Lipschitz nonlinearity. We completely characterize one-dimensional bounded solutions of this problem, and we prove among other things that if $u$ is a bounded solution with $ρ:=\sup_{\mathbb{R}^N}u$ verifying $f(ρ)=0$, then $u$ is necessarily one-dimensional.

math.AP↗

arXiv2015

A priori bounds and existence of solutions for some nonlocal elliptic problems

In this paper we show existence of solutions for some elliptic problems with nonlocal diffusion by means of nonvariational tools. Our proof is based on the use of topological degree, which requires a priori bounds for the solutions. We obtain the a priori bounds by adapting the classical scaling method of Gidas and Spruck. We also deal with problems involving gradient terms.

math.AP↗