Non-Variational Elliptic Systems in Dimension Two: A Priori Bounds and Existence of Positive Solutions

Djairo G. de Figueiredo (), João Marcos do Ó () and Bernhard Ruf ()
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Djairo G. de Figueiredo: Universidade Estadual de Campinas, IMECC–Departamento de Matemática
João Marcos do Ó: Universidade Federal da Paraíba, CCEN–Departamento de Matemática
Bernhard Ruf: Universitàdegli Studi, Dipartimento di Matematica

A chapter in Djairo G. de Figueiredo - Selected Papers, 2008, pp 661-680 from Springer

Abstract: Abstract We establish a priori bounds for positive solutions of semilinear elliptic systems of the form $$ \left\{ {\begin{array}{*{20}c} { - \Updelta u = g(x,u,v)} & {{\text{in}}\;\;\Upomega ,} \\ { - \Updelta v = f(x,u,v)} & {{\text{in}}\;\;\Upomega ,} \\ {u\; > \;0,\quad v\; > \;0} & {{\text{in}}\;\;\Upomega ,} \\ {u = v = 0} & {{\text{on}}\;\;\partial \Upomega ,} \\ \end{array} } \right. $$ where Ω is a bounded and smooth domain in ℝ2. We obtain results concerning such bounds when f and g depend exponentially on u and v. Based on these bounds, existence of positive solutions is proved.

Keywords: Elliptic systems; A priori estimates; 35J60; 35B45 (search for similar items in EconPapers)
Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-02856-9_41

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DOI: 10.1007/978-3-319-02856-9_41

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