Degenerate semiconductor device equations with temperature effect

In this paper, we establish an existence assertion for the following elliptic system:-div(a(v)∇n-a(v)n∇ψ)=0 in Ω,-div(a(v)∇p+a(v)p∇ψ)=0 in Ω,-div(a(v)∇ψ)=p-n+f in Ω,-div(a(v)∇v)=a(v)|∇ψ|2 in Ωcoupled with suitable boundary conditions. This problem arises from the study of semiconductor devices with temperature effect and without recombination. We only assume that a   is continuous and positive. This gives rise to the possibility that the system may be degenerate and/or singular. We show that, if a(s)a(s) does not go to zero too fast as s→∞s→∞, there exists a bounded weak solution, and therefore neither degeneracy nor singularity really occur. This also immediately implies some additional regularity properties for the weak solution.