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Marta Garcia-Huidobro

Publications and source records attributed to Marta Garcia-Huidobro.

At least 19 recordsLinked to original sources
arXiv2026

Local and global properties of solutions of an elliptic equation involving exponential and gradient reaction

We study some local and global properties of solutions of $-Δu- m\abs{\nabla u}^q-e^{u}=0$ in a punctured domain $Ω\setminus\{0\}$, or in an exterior domain of $R^N$, $N\geq 2$, where $m$ is a positive parameter and $q>1$. We study particularly the local behaviour of solutions with an isolated singularity or the asymptotic behaviour for solutions defined in an exterior domain, and also the existence of solutions with the behaviours previously described. These behaviours change drastically according $q$ is smaller or larger than $2$. Many results are obtained by introducing various dynamical systems associated to the equation.

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arXiv2022

Boundary singular solutions of a class of equations with mixed absorption-reaction

We study properties of positive functions satisfying (E) --$Δ$u + u p -- M |$\nabla$u| q = 0 is a domain $Ω$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the boundary except at one point. This analysis depends on the existence of separable solutions in R N +. We consruct various types of positive solutions with an isolated singularity on the boundary. We also study conditions for the removability of compact boundary sets and the Dirichlet problem associated to (E) with a measure for boundary data.

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arXiv2021

Measure data problems for a class of elliptic equations with mixed absorption-reaction

We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-Δu+u^p-M|\nabla u|^q=μ$ in a domain $Ω\subset\BBR^N$ where $μ$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $Ω\setminus K$ where $K$ is a compact subset of $Ω$ can be extended as a solution of the same equation in $Ω$

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arXiv2020

On the uniqueness of solutions of a semilinear equation in an annulus

We establish the uniqueness of positive radial solutions of $$\begin{cases} Δu +f(u)=0,\quad x\in A \\ u(x) =0 \quad x\in \partial A \end{cases} $$ where $A:=A_{a,b}=\{ x\in {\mathbb R}^n : a<|x| 0$ for all $s>0$, or has one zero at $B>0$, is non positive and not identically 0 in $(0,B)$ and it is positive in $(B,\infty)$.

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arXiv2019

On the uniqueness of bound state solutions of a semilinear equation with weights

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0 0$, is non positive and not identically 0 in $(0,b)$, positive in $(b,c)$, and is differentiable in $(0,c)$.

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arXiv2019

Interpolation inequalities in W1,p(S1) and carr{é} du champ methods

This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carr{é} du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in W1,p(S1) with p $\ge$ 2. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a p-Laplacian type operator. It is remarkable that the carr{é} du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever p$\ne$2.

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arXiv2018

Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

We study local and global properties of positive solutions of $-Δu=u^p]{\left |{\nabla u}\right |}^q$ in a domain $Ω$ of ${\mathbb R}^N$, in the range $1\<p+q$, $p\geq 0$, $0\leq q\< 2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\mathbb R}^N$ when $p(N-2)+q(N-1) \<N$ or in an exterior domain if $p(N-2)+q(N-1)\<N$ and $0\leq q\<1$. Using a direct Bernstein method we obtain a first range of values of $p$ and $q$ in which $u(x)\leq c({\mathrm dist\,}(x,\partialΩ)^{\frac{q-2}{p+q-1}}$ This holds in particular if $p+q\<1+\frac{4}{n-1}$. Using an integral Bernstein method we obtain a wider range of values of $p$ and $q$ in which all the global solutions are constants. Our result contains Gidas and Spruck nonexistence result as a particular case. We also study solutions under the form $u(x)=r^{\frac{q-2}{p+q-1}}ω(σ)$. We prove existence, nonexistence and rigidity of the spherical component $ω$ in some range of values of $N$, $p$ and $q$.

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arXiv2018

Separable infinite harmonic functions in cones

We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $σ$) = r --$β$ $ω$($σ$). We prove that such solutions exist, the spherical part $ω$ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N --1 and that the exponents $β$ = $β$ + > 0 and $β$ = $β$ -- < 0 are uniquely determined if the domain is smooth.

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arXiv2015

Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations

We study the boundary behaviour of the solutions of (E) $-Δ_p u+|\nabla u|^q=0$ in a domain $Ω\subset \mathbb{R}^N$, when $N\geq p > q >p-1$. We show the existence of a critical exponent $q_* < p$ such that if $p-1 < q < q_*$ there exist positive solutions of (E) with an isolated singularity on $\partialΩ$ and that these solutions belong to two different classes of singular solutions. If $q_*\leq q < p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.

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arXiv2014

Existence of sign changing solutions for an equation with a weighted p-Laplace operator

We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

math.AP↗
arXiv2014

Local and global properties of solutions of quasilinear Hamilton-Jacobi equations

We study some properties of the solutions of (E) $\;-\Gd_p u+|\nabla u|^q=0$ in a domain $\Gw \sbs \BBR^N$, mostly when $p\geq q>p-1$. We give a universal priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the positive solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result in expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete non compact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.

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arXiv2013

Keller-Osserman estimates for some quasilinear elliptic systems

In this article we study quasilinear multipower systems of two equations of two types, in a domain $Ω$ of R^{N} : with absorption terms, or mixed terms. Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type. Concerning the mixed system, we show that one of the solutions always satisfies Harnack inequality. In the case $Ω$=B(0,1)\{0}, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.

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arXiv2013

Quasilinear elliptic Hamilton-Jacobi equations on complete manifolds

Let (M^n,g) be a n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Ricc_g and sectional curvature Sec_g. Assume Ricc_g\geq (1-n)B^2, and either p>2 and Sec_g(x)=o(dist^2(x,a)) when dist^2(x,a)\to\infty for a\in M, or 1 p-1> 0, any C^1 solution of (E) -\Gd_pu+\abs{\nabla u}^q=0 on M satisfies \abs{\nabla u(x)}\leq c_{n,p,q}B^{\frac{1}{q+1-p}} for some constant c_{n,p,q}>0. As a consequence there exists c_{n,p}>0 such that any positive p-harmonic function v on M satisfies v(a)e^{-c_{n,p}B\dist (x,a)}\leq v(x)\leq v(a)e^{c_{n,p}B\dist (x,a)} for any (a,x)\in M\times M.

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arXiv2013

Remarks on some quasilinear equations with gradient terms and measure data

Let $Ω\subset \mathbb{R}^{N}$ be a smooth bounded domain, $H$ a Caratheodory function defined in $Ω\times \mathbb{R\times R}^{N},$ and $μ$ a bounded Radon measure in $Ω.$ We study the problem% \begin{equation*} -Δ_{p}u+H(x,u,\nabla u)=μ\quad \text{in}Ω,\qquad u=0\quad \text{on}\partial Ω, \end{equation*} where $Δ_{p}$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm \left\| \nabla u\right\| ^{q}$ ($q>0$). We obtain an existence result under subcritical growth assumptions on $H,$ we give necessary conditions of existence in terms of capacity properties, and we prove removability results of eventual singularities. In the supercritical case, when $μ\geqq 0$ and $H$ is an absorption term, i.e. $% H\geqq 0,$ we give two sufficient conditions for existence of a nonnegative solution.

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arXiv2012

On the existence of sign changing bound state solutions of a quasilinear equation

In this paper we establish the existence of bound state solutions of any given order to $$ Δ_m u +f(u)=0, x\in R^N, N\ge m>1, (P) $$ where $Δ_m u=\nabla\cdot(|\nabla u|^{m-2}\nabla u)$ using the same techniques as in [GST] to establish the existence of a ground state solution to (P). Since our solutions change sign, we assume f is continuous in R. The main point here is that by asking a stronger subcritical assumption (see (f_3)(ii) below) than the one considered in [GST], we are able to adapt their techniques to obtain the existence of bound states with a prescribed number of zeros.

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