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Alexander Quaas

Publications and source records attributed to Alexander Quaas.

At least 37 records · Page 2Linked to original sources
arXiv2021

A note on one-dimensional symmetry for Hamilton-Jacobi equations with extremal Pucci operators and application to Bernstein type estimate

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative $L^q$-viscosity solutions of the equation \begin{equation*} -\mathcal{M}_{λ, Λ}^{\pm}(D^2u)\pm |Du|^p=0, x\in \mathbb{R}_+^n, \end{equation*} with boundary condition $u(\tilde{x},0)=M\geq 0, \tilde{x}\in \mathbb{R}^{n-1}$, where $\mathcal{M}_{λ, Λ}^{\pm}$ are the Pucci's operators with parameters $λ, Λ\in \mathbb{R}_+$ $0<λ\leq Λ$ and $p>1$. The results are an extension of the results by Porreta and Verón in arXiv:0805.2533 for the case $p\in (1,2]$ and by o Filippucci, Pucci and Souplet in arXiv:1906.05161 for the case $p>2$, both for the Laplacian case (i.e. $λ=Λ=1$). As an application in the case $p>2$, we prove a sharp Bernstein estimation for $L^q$-viscosity solutions of the fully nonlinear equation \begin{equation*} -\mathcal{M}_{λ, Λ}^{\pm}(D^2u)= |Du|^p+f(x), \quad x\in Ω, \label{ecuacion1} \end{equation*} with boundary condition $u=0$ on $\partial Ω$, where $Ω\subset \mathbb{R}^n$.

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arXiv2020

Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when $t\to +\infty$, i.e., they behave like $λ^*t + ϕ(x)$ where $λ^*$ is the maximal ergodic constant and $ϕ$ is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem.

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arXiv2019

Spectrum of the fractional $p-$Laplacian in $\mathbb{R}^N$ and decay estimate for positive solutions of a Schrödinger equation

In this paper, we prove the existence of unbounded sequence of eigenvalues for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a nonexistence result when the weighthas positive integral. In addition, we show some qualitative properties of the first eigenfunction including a sharp decay estimate. Finally, we extend the decay result to the positive solutions of a Schrödinger type equation.

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arXiv2018

On nonhomogeneous elliptic equations with the Hardy-Leray potentials

In this paper, we present some suitable distributional identities of the solutions for nonhomogeneous elliptic equations involving the Hardy-Leray potentials and study qualitative properties of the solutions to the corresponding nonhomogeneous problems by the distributional identities. We address some applications on the nonexistence of some nonhomogeneous problems with the Hardy-Leray potentials and the nonexistence principle eigenvalue with some indefinite potentials.

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arXiv2018

Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity solutions. The Dirichlet problem is well posed globally in time when boundary data is assumed to be satisfied in the latter sense. Thus, our main results are \emph{a)} the existence of solutions which satisfy the boundary data in the classical sense for a small time, for all Hölder-continuous initial data, with Hölder exponent above a critical a value, and \emph{b)} the nonexistence of solutions satisfying the boundary data in the classical sense for all time. In this case, the phenomenon of loss of boundary conditions occurs in finite time, depending on a largeness condition on the initial data.

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arXiv2017

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, that a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.

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arXiv2016

Global bifurcation for fractional $p$-Laplacian and application

We prove the existence of an unbounded branch of solutions to the non-linear non-local equation $$ (-Δ)^s_p u=λ|u|^{p-2}u + f(x,u,λ) \quad\text{in}\quad Ω,\quad u=0 \quad\text{in}\quad \mathbb{R}^n\setminusΩ, $$ bifurcating from the first eigenvalue. Here $(-Δ)^s_p$ denotes the fractional $p$-Laplacian and $Ω\subset\mathbb{R}^n$ is a bounded regular domain. The proof of the bifurcation results relies in computing the Leray--Schauder degree by making an homotopy respect to $s$ (the order of the fractional $p$-Laplacian) and then to use results of local case (that is $s=1$) found in [17]. Finally, we give some application to an existence result.

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arXiv2016

Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions

In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu= a_λ(x)|u|^{q-2}u+b(x)|u|^{2^*_α-1}u &{\rm in}\,\,Ω, u=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $0<α<1$, $ Ω$ is a bounded domain with smooth boundary in $ \R^N $ with $N>2α$ and $ 2^*_α=2N/(N-2α)$ is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Ljusternik-Schnirelmann category.

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arXiv2015

Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in R^N involving fractional Laplacian

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.

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arXiv2015

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=f(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ (-Δ)^αv=g(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ u=v=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $(-Δ)^α$ denotes the fractional Laplacian and $ Ω$ is a smooth bounded domain in $ \R^N $. It shown that under some assumptions on $ f $ and $ g $, the problem has at least one positive solution $(u,v)$. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

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arXiv2015

A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian

We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=v^q&{\rm in}\,\,\R^N,\\ (-Δ)^αv=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eqnarray*} where $ α\in(0,1) $, $ N>2α$ and $ p,q $ are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre \cite{CS}. Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in an infinity half cylinder based on some maximum principles which obtained by some barrier functions and a coupling argument using fractional Sobolev trace inequality.

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arXiv2015

Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^αu=u^p\quad &{\rm in}\quad Ω\setminus\{0\},\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminusΩ, \end{array} \end{equation} where $p>1$, $Ω$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing the origin, $N\ge2$ and the fractional Laplacian $(-Δ)^α$ is defined in the principle value sense. We obtain that any classical solution $u$ of (\ref{eq 0.1}) is a weak solution of \begin{equation}\label{eq 0.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^αu=u^p+kδ_0\quad &{\rm in}\quad Ω,\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminusΩ\end{array} \end{equation} for some $k\ge0$, where $δ_0$ is the Dirac mass at the origin. In particular, when $p\ge \frac{N}{N-2α}$, we have that $k=0$; when $p< \frac{N}{N-2α}$, $u$ has removable singularity at the origin if $k=0$ and if $k>0$, $u$ satisfies $$\lim_{x\to0} u(x)|x|^{N-2α}=c_{N,α}k,$$ where $c_{N,α}>0$. Furthermore, when $p\in(1, \frac{N}{N-2α})$, we obtain that there exists $k^*>0$ such that problem (\ref{eq 0.1}) has at least two positive solutions for $k k^*$.

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arXiv2013

Self-generated interior blow-up solutions in fractional elliptic equation with absorption

In this paper we study positive solutions to problem involving the fractional Laplacian $(E)$ $(-Δ)^α u(x)+|u|^{p-1}u(x)=0 in x\inΩ\setminus\mathcal{C}$, subject to the conditions $u(x)=0$ $x\inΩ^c$ and $\lim_{x\inΩ\setminus\mathcal{C}, x\to\mathcal{C}}u(x)=+\infty$, where $p>1$ and $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $\mathcal{C}\subset Ω$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem $(E)$. Moreover, we further analyze uniqueness, asymptotic behaviour and nonexistence.

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