arXiv subjects

Cecilia Yarur

Publications and source records attributed to Cecilia Yarur.

6 recordsLinked to original sources

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.

math.AP↗

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.

math.AP↗

arXiv2011

Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion

In this article we study the positive solutions of the parabolic semilinear system of competitive type \[ \left\{\begin{array} [c]{c}% u_{t}-Δu+v^{p}=0, v_{t}-Δv+u^{q}=0, \end{array} \right. \] in $Ω\times\left(0,T\right) $, where $Ω$ is a domain of $\mathbb{R}^{N},$ and $p,q>0,$ $pq\neq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \[ u(x,t)\leqq Ct^{-(p+1)/(pq-1)},\qquad v(x,t)\leqq Ct^{-(q+1)/(pq-1)}% \] in $ω\times\left(0,T_{1}\right) ,$ for any domain $ω\subset\subsetΩ$ and $T_{1}\in\left(0,T\right) ,$ and $C=C(N,p,q,T_{1}% ,ω).$ For $p,q>1,$ we prove the existence of an initial trace at time 0, which is a Borel measure on $Ω.$ Finally we prove that the punctual singularities at time $0$ are removable when $p,q\geqq1+2/N.

math.AP↗

arXiv2010

Large solutions of elliptic systems of second order and applications to the biharmonic equation

In this work we study the nonnegative solutions of the elliptic system Δu=|x|^{a}v^δ, Δv=|x|^{b}u^μ in the superlinear case μδ>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ^2 u=|x|^{b}|u|^μ. Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, combined with nonradial upper estimates.

math.AP↗

arXiv2010

Boundary value problems with measures for elliptic equations with singular potentials

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-Δu+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure $\gm$ on $\prt\Gw$ so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In the appendix A. Ancona solves a question raised by M. Marcus and L. Véron concerning the vanishing set of the Poisson kernel of $L_V$ for an important class of potentials $V$.

math.AP↗

arXiv2010

On the uniqueness of sign changing bound state solutions of a semilinear equation

We establish the uniqueness of the higher radial bound state solutions of $$ Δu +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth conditions, and either has one zero at $b>0$, is non positive and not identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or is positive and differentiable in $[0,\infty)$.

math.AP↗