2013 | OriginalPaper | Chapter
A Maximum Principle for an Elliptic System and Applications to Semilinear Problems
Authors : Djairo G. de Figueiredot, Enzo Mitidieri
Published in: Djairo G. de Figueiredo - Selected Papers
Publisher: Springer International Publishing
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The Dirichlet problem in a bounded region for elliptic systems of the form (*)
$$ - \Updelta u = f\left( {x,u} \right) - v,\quad - \Updelta v = \delta u - \gamma v $$
is studied. For the question of existence of positive solutions the key ingredient is a maximum principle for a linear elliptic system associated with (*). A priori bounds for the solutions of (*) are proved under various types of growth conditions on f. Variational methods are used to establish the existence of pairs of solutions for (*).