Some existence results for a nonlinear problem with source term in Zygmund space

Abstract

In this paper we study the existence of solutions to the nonlinear elliptic problem in the limit case $N\ge 3$, $u\in W_0^{1,N}(\Omega )$, $-\mathrm{div}a(x,u,Du)=H(x,u,Du)+a_0(x)|u{|}^{N-2}u+f(x)$ in ${\mathcal{D}}^{\prime }(\Omega )$, where $\Omega$ is bounded open subset of ${\mathbb{R}}^N(N\ge 3)$, $-\mathrm{div}a(x,u,Du)$ is a Leray–Lions operator defined on $W_0^{1,N}(\Omega )$, $H(x,s,\xi )$ is a Carathéodory function which satisfies for some $\gamma >0$ and $c_0\ge 0$, $-c_{0}a(x,s,\xi )\xi \le H(x,s,\xi )\mathrm{sign}(s)\le \gamma a(x,s,\xi )\xi$. Here, $a_0(x)\ge 0$ belongs to $L^q(\Omega )$ with $q>1$, and $f(x)$ belongs to $L^1(\log L{)}^{(N-1)}(\Omega )$. For $f$ and $a_0$ sufficiently small, we prove the existence of at least one solution $u$ of this problem, which is such that $e^{{\mu }_0|u|}-1$ belongs to $W_0^{1,N}$ for some ${\mu }_0\ge \gamma$ and satisfies an a priori estimate.