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On maximum principles and existence of positive solutions for some cooperative elliptic systems. (English) Zbl 0821.35018

This paper is concerned with the Dirichlet zero boundary value problems for some cooperative quasilinear elliptic systems involving the \(p\)- Laplacian on a smooth bounded domain in \(\mathbb{R}^ d\). The authors give alternative simple proofs for some of the results for \(p=2\) in D. G. de Figueiredo and E. Mitidieri [Rend. Ist. Mat. Univ. Trieste 22, 36-66 (1990; Zbl 0793.35011)] on maximum principle and existence of positive solutions, and they prove existence and uniqueness of positive solutions for some semilinear elliptic systems which are bounded nonlinear perturbations for cooperative linear systems. Moreover, they extend some of these results to the \(p\)-Laplacian for \(p\neq 2\). Related results for linear and semilinear cooperative elliptic systems are given in M. A. S. Souto [Differ. Integral Equ. 8, 1245-1258 (1995)]. Also, the existence and uniqueness of positive solutions to some strictly cooperative quasilinear elliptic systems involving the \(p\)-Laplacian are shown in J. Fleckinger-Pellé and P. Takáč [Indiana Univ. Math. J. 43, 1227-1253 (1994)].

MSC:

35B50 Maximum principles in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J30 Higher-order elliptic equations

Citations:

Zbl 0793.35011