An elliptic equation with power nonlinearity and degenerate coercivity

Abstract

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