A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution u : Ω → R to the nonlinear elliptic partial differential equation − ∇ ⋅ ( a ( u ( x ) ) ∇ u ( x ) ) = f ( x ) under periodic boundary conditions, where u ( x 0 ) = u 0 at x 0 ∈ Ω , with Ω = T N , the N-dimensional torus, and N = 2 , 3 . The function a is assumed to be smooth, and a ( u ( x ) ) > 0 for u ( x ) ∈ G ¯ , where G ⊂ R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval G 1 ¯ ⊂ G ¯ . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x 0 .