An elliptic equation with power nonlinearity and degenerate coercivity

Abstract

We discuss the existence and regularity of solutions to the following Dirichlet problem: $$\begin{equation} \begin{cases} -\textrm{div}\left(\frac{Du}{(1+|u|)^θ}\right)= -\textrm{div}\left(u^γE(x)\right)+f(x) \qquad & \mbox{in } Ω,\\ u (x) = 0 & \mbox{on } \partial Ω, \end{cases} \end{equation}$$ where $\hat{I}¸,{\hat{I}}^3>0$. An interesting feature of this problem is the interplay between the two nonlinearities, the degeneracy and the power nonlinearity.