On some weakly coercive quasilinear problems with forcing

TL;DR

This paper proves the existence of solutions for a class of weakly coercive quasilinear PDEs involving the p-Laplacian and potential functions, including Hardy and Poincaré potentials, under certain conditions.

Contribution

It establishes solution existence for a broad class of weakly coercive quasilinear problems with specific potential functions, extending previous results.

Findings

Solutions exist for all distributional forcing functions under the given conditions.

Includes cases with Hardy potential and Poincaré constant potential.

Provides a framework for analyzing weakly coercive quasilinear PDEs.

Abstract

We consider the forced problem $-\Delta_p u - V(x)|u|^{p-2} u = f(x)$, where $\Delta_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $\Omega\subset \mathbb{R}^N$, $V\ge 0$ and $Q_V (u) := \int_\Omega |\nabla u|^p\, dx - \int_\Omega V|u|^p\,dx$ satisfies the condition (A) stated at the beginning of the paper. We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)=\lambda_{1,p}(\Omega)$, where $\lambda_{1,p}(\Omega)$ is the Poincar\'e constant on an infinite strip.