An elliptic equation with power nonlinearity and degenerate coercivity

TL;DR

This paper investigates the existence and regularity of solutions to a nonlinear elliptic PDE with degenerate coercivity and power nonlinearity, highlighting the complex interaction between these nonlinear features.

Contribution

It provides new results on solution existence and regularity for a class of elliptic equations with combined degeneracy and power nonlinearities.

Findings

Existence of solutions under certain conditions.

Regularity results for solutions.

Analysis of the interplay between degeneracy and nonlinearity.

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.