de Figueiredo, Djairo G.; Mitidieri, Enzo A maximum principle for an elliptic system and applications to semilinear problems. (English) Zbl 0608.35022 SIAM J. Math. Anal. 17, 836-849 (1986). For a given function u, let Bu denote the solution of the Dirichlet problem \(-\Delta v+\gamma v=\delta u\) in \(\Omega\), \(v=0\) on \(\partial \Omega\), where \(\Omega\) is a bounded domain in \(R^ N\), \(N\geq 2\), with smooth boundary \(\partial \Omega\) and where \(\gamma\) and \(\delta\) are positive constants. Then in order to find u which solves \[ -\Delta u+Bu=f(x,u)\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega, \] the authors study the spectral properties of the operator \(-\Delta +B\), establish a maximum principle for solutions of the problem \(-\Delta u+Bu-\lambda u=g(x)\) in \(\Omega,u=0\) on \(\partial \Omega\), where \(\lambda\) is restricted to certain ranges depending on \(\gamma\),\(\delta\), and the domain \(\Omega\), deduce a priori bounds for a sublinear elliptic system, and use variational methods to establish the existence of solutions to the system involving u and v. These solutions represent steady state solutions of reaction-diffusion systems of interest in biology. Reviewer: P.Schaefer Cited in 73 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles in context of PDEs 35A15 Variational methods applied to PDEs 35B45 A priori estimates in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 47J05 Equations involving nonlinear operators (general) 35A35 Theoretical approximation in context of PDEs 35J50 Variational methods for elliptic systems Keywords:semilinear problems; positive solutions; monotone iteration; mountain pass theorem; Dirichlet problem; maximum principle; a priori bounds; sublinear elliptic system; variational methods; existence; steady state; reaction-diffusion systems PDFBibTeX XMLCite \textit{D. G. de Figueiredo} and \textit{E. Mitidieri}, SIAM J. Math. Anal. 17, 836--849 (1986; Zbl 0608.35022) Full Text: DOI