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A maximum principle for an elliptic system and applications to semilinear problems. (English) Zbl 0608.35022

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

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
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