A nonlinear degenerate elliptic problem with W^{1,1}_0 solutions

TL;DR

This paper investigates a nonlinear elliptic problem with degenerate coercivity, establishing existence and uniqueness of solutions in W^{1,1}_0 spaces under certain conditions, and demonstrating non-existence for measures on sets of zero harmonic capacity.

Contribution

It provides the first existence and uniqueness results for W^{1,1}_0 solutions to such degenerate elliptic problems and characterizes non-existence cases involving measures.

Findings

Existence of unique W^{1,1}_0 solutions under summability assumptions.

Non-existence of solutions when the source is a Radon measure on sets of zero harmonic capacity.

Characterization of solution behavior in degenerate elliptic problems.

Abstract

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.