On a coupled system of partial differential equations (Q1095306)

We investigate the following coupled system of stationary partial differential equations: (1) \[ \nabla \cdot (D_ i\nabla u_ i)+\nabla \cdot (k_ iz_ iu_ i(a+bu^ 2_ 0)^{-}\nabla \psi)=f_ i;\quad -\nabla \cdot (D_ 0\nabla \psi)=k(N+u_ 0),\quad i=1,...,M \] where \(u_ i\), \(f_ i\), \(\psi\) and N are real functions on a bounded domain G of the n-dimensional Euclidean space \({\mathbb{R}}_ n\) allowing partial integration and the Sobolev imbedding theorems. \(\nabla \{D\nabla u)\) means div(D grad u). Let further \(u_ 0=z_ 1u_ 1+...+z_ Mu_ M\), \(z_ i=\pm 1\), \(U_ 0=1/(a+bu^ 2_ 0)^{1/2}\), M a natural number, k, \(k_ i\), a, b positive constants. The diffusion coefficients may be of the form \(D_ i=D_ i(| \nabla u_ i|)\) and \(D_ 0=D_ 0(| \nabla \psi |)\). For all functions \(u_ i\) and \(\psi\) we suppose the same type of boundary conditions: (2) \[ u_ i|_{\partial G_ 1}=\underline y_ i,\quad i=1,...,M,\quad \psi |_{\partial G_ 1}=\underline y_ 0;\quad J_ i\cdot \vec n|_{\partial G_ 2}=0;\quad i=0,1,...,M \] where \(J_ 0=-D_ 0\nabla \psi\) and \(J_ i=-D_ i\nabla u_ i-k_ iz_ iu_ i(a+bu_ 0^ 2)^{-}\nabla \psi\) are the flow densities, \(\partial G_ 1\) and \(\partial G_ 2\) are portions of the boundary of G with the properties \(\partial G_ 1\cap \partial G_ 2=\emptyset\), \(\partial G=\partial G_ 1\cup \partial G_ 2\), mes \(\partial G_ 1>0\) with respect to the boundary measure, and \(\vec n\) is outer normal to the boundary of G. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.