$W^{1,1}_{0}({\varOmega })$ solutions for some nonlinear elliptic equations
Volume 167 / 2022
Colloquium Mathematicum 167 (2022), 115-126
MSC: Primary 35A01, 35J60, 35J66, 35J92.
DOI: 10.4064/cm8364-12-2020
Published online: 12 May 2021
Abstract
We prove the existence of renormalized solutions belonging to $W^{1,1}_{0}(\Omega )$ of a class of strongly nonlinear elliptic $p$-Laplace type problems when $1 \lt p \lt 2-1/N$ with data of poor summability in the Lebesgue space $L^{m}(\Omega )$, $m=N/((p-1)N+1)$. Our method consists in applying Schaefer’s classical fixed point theorem relying on new estimates of the gradient of the unique renormalized solution of problems with datum in $L^{m}(\Omega )$.
