Existence of Solutions for a Nonlinear Elliptic Equation with General Flux Term

which contains the flux term J. The flux term J is a vector field that explains a movement of some physical contents u such as temperature, chemical potential, electrostatic potential, or fluid flows. Physical observations tell us in general that J depends on u and approximately on its gradient at each point x, that is, J J x,∇, u x . For linear cases, one can simply represent J as J c∇u on isotropic medium or J A∇u with a square matrix A on an isotropic medium . But for nonlinear cases, the situation can be much more complicated. One of the common assumptions is that J |∇u|p−2∇u is to produce the p-Laplacian