Searcharxiv⌕ Search

arXiv subjects

Eduardo Colorado

Publications and source records attributed to Eduardo Colorado.

12 recordsLinked to original sources
arXiv2025

Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

We consider the following nonlocal critical problem with mixed Dirichlet-Neumann boundary conditions, \begin{equation} \left\{ \begin{array}{ll} (-Δ)^su=λu+|u|^{2_s^*-2}u &\text{in}\ Ω,\\ \mkern+38.5mu u=0& \text{on}\ Σ_{\mathcal{D}},\\ \mkern+24mu \displaystyle \frac{\partial u}{\partial ν}=0 &\text{on}\ Σ_{\mathcal{N}}, \end{array} \right. \end{equation} where $(-Δ)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $Ω\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $2_s^*=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $λ>0$ is a real parameter, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$--dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$ and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$--dimensional submanifold of $\partialΩ$. By employing a $\nabla$-theorem we prove the existence of multiple solutions when the parameter $λ$ is in a left neighborhood of a given eigenvalue of $(-Δ)^s$.

math.AP↗
arXiv2021

Nonlinear Fractional Schrödinger Equations coupled by power-type nonlinearities

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schrödinger equations, \begin{equation*} \left \{ \begin{array}{l} (-Δ)^s u_1+ λ_1 u_1= μ_1 |u_1|^{2p-2}u_1+β|u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N,\\[3pt] (-Δ)^s u_2 + λ_2 u_2= μ_2 |u_2|^{2p-2}u_2+β|u_1|^{p}|u_2|^{p-2}u_2 \quad\text{in }\mathbb{R}^N, \end{array} \right. \end{equation*} where $ u_1,\, u_2\in W^{s,2}(\mathbb{R}^N)$, with $ N=1,\, 2,\, 3$; $λ_j,\,μ_j>0$, $j=1,2$, $β\in \mathbb{R}$, $p\geq 2$ and $\displaystyle\frac{p-1}{2p}N 0$ for $j=1,\ldots ,m\ge 3$, the coupling parameters $β_{jk}=β_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,m$, $j\neq k$. For this system we prove similar results as for $m=2$, depending on the values of the parameters $β_{jk}, p, λ_j,μ_j$, (for $j,k=1,\ldots,m$, $j\neq k$).

math.AP↗
arXiv2021

Bound and ground states of coupled "NLS-KDV" equations with Hardy potential and critical power

We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call ``Schrödinger-Korteweg-de Vries'' non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter $ν$. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.

math.AP↗
arXiv2021

Existence of bound and ground states for an elliptic system with double criticality

We study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter $ν$ is large or $ν$ is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain-Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves some known results and allows us to cover range of parameters which have not been considered previously.

math.AP↗
arXiv2019

Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator

This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, $$(-Δ)^α u=λu+ (-Δ)^β|u|^{p-1}u \quad \mbox{in}\quad Ω;\qquad (-Δ)^{j}u=0\quad \mbox{on}\quad \partialΩ,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [α]$$ where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $0<β<1$, $β<α<β+1$ and $λ>0$. In particular, we study the fractional elliptic problem, $$ (-Δ)^{α-β} u= λ(-Δ)^{-β}u+ |u|^{p-1}u \quad\mbox{in} \quad Ω;\qquad u=0 \quad \hbox{on} \quad \partialΩ,$$ and we prove existence or nonexistence of positive solutions depending on the parameter $λ>0$, up to the critical value of the exponent $p$, i.e., for $1<p\leq 2_μ^*-1$ where $μ:=α-β$ and $2_μ^*=\frac{2N}{N-2μ}$ is the critical exponent of the Sobolev embedding.

math.AP↗
arXiv2019

Existence of positive solutions for a Brezis--Nirenberg type problem involving an inverse operator

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-Δ)^2 u=λu+ (-Δ)|u|^{p-1}u,$$ in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-Δu=λ(-Δ)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $λ>0$, up to the critical value of the exponent $p$, i.e., when $1<p\leq 2^*-1$, where $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent. For $p=2^*-1$, this equivalence leads us to a Brezis--Nirenberg type problem, cf. \cite{BN}, but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant ensures the existence of solution, going from dimensions $N\geq 4$ in the classical Brezis-Nirenberg problem, to dimensions $N\geq7$ for this nonlocal problem.

math.AP↗
arXiv2018

The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions

In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \mathbb{R}$, $ν$ is the outwards normal to $\partialΩ$, $Σ_{\mathcal{D}}$, $Σ_{\mathcal{N}}$ are smooth $(N-1)$-dimensional submanifolds of $\partialΩ$ such that $Σ_{\mathcal{D}}\cupΣ_{\mathcal{N}}=\partialΩ$, $Σ_{\mathcal{D}}\capΣ_{\mathcal{N}}=\emptyset$, and $Σ_{\mathcal{D}}\cap\overlineΣ_{\mathcal{N}}=Γ$ is a smooth $(N-2)$-dimensional submanifold of $\partialΩ$.

math.AP↗
arXiv2018

Ground states of some coupled nonlocal fractional dispersive PDEs

We show the existence of ground state solutions to the following stationary system coming from some coupled fractional dispersive equations such as: nonlinear fractional Schrödinger (NLFS) equations (for dimension $n=1,\, 2,\, 3$) or NLFS and fractional Korteweg-de Vries equations (for $n=1$), $$ \left \{ \begin{array}{ll} (-Δ)^{s} u+ λ_1 u &= u_1^{3}+βuv,\quad u\in W^{s,2}(\mathbb{R}^n), (-Δ)^{s} v + λ_2 v &= \frac 12 v^{2}+\frac 12 βu^2,\quad v\in W^{s,2}(\mathbb{R}^n), \end{array} \right. $$ where $λ_j>0$, $j=1,2$, $β\in \mathbb{R}$, $n=1,\, 2,\, 3$, and $\frac n4< s<1$. Precisely, we prove the existence of a positive radially symmetric ground state for any $β>0$.

math.AP↗
arXiv2014

On the existence of bound and ground states for some coupled nonlinear Schrödinger--Korteweg-de Vries equations

We demonstrate existence of positive bound and ground states for a system of coupled nonlinear Schrödinger--Korteweg-de Vries equations. More precisely, we prove there is a positive radially symmetric ground state if either the coupling coefficient $β>Λ$ (for an appropriate constant $Λ>0$) or $β>0$ with appropriate conditions on the other parameters of the problem. Concerning bound states, we prove there exists a positive radially symmetric bound state if either $0<β$ is sufficiently small or $0<β<Λ$ with some appropriate conditions on the parameters as for the ground states. That results give a classification of positive solutions as well as multiplicity of positive solutions. Furthermore, we consider a system with more general power nonlinearities, proving the above results, and also we study natural extended systems with more than two equations. Although the techniques we employed are variational, we look for critical points of an energy functional different from the classical one used in this kind of systems. Our approach improves many of the previous known results, as well as permit us to show new results not considered or studied before.

math.AP↗
arXiv2014

Positive solutions to some systems of coupled nonlinear Schrödinger equations

We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schrödinger equations $$ - Δu_j+ λ_j u_j =μ_j u_j^3+ \sum_{k=1;k\neq j}^Nβ_{jk} u_ju_k^2,\quad u_j\in W^{1,2}(\mathbb{R}^n);\: j=1,\ldots,N $$ where $n=1,\,2,\, 3; \,λ_j,\, μ_j>0$ for $j=1,\ldots,N$, the coupling parameters $β_{jk}=β_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,N$, $j\neq k$. Precisely, we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors. Additionally, with more restrictive conditions on the coupled parameters, we show that the bound states founded are positive.

math.AP↗
arXiv2010

A concave-convex elliptic problem involving the fractional Laplacian

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the problem exist and prove a multiplicity result.

math.AP↗