Convergence of weak solutions of elliptic problems with datum in $L^1$

Abstract

Motivated by the $Q$-condition result proven by Arcoya and Boccardo in [J. Funct. Anal. 268(2015), No. 5, 1153–1166], we analyze the behaviour of the weak solutions $\{u_\varepsilon\}$ of the problems \begin{equation*} \begin{cases} -\Delta_p u_\varepsilon + \varepsilon|f(x)|u_\varepsilon = f(x) &\text{in } \Omega,\\ u_\varepsilon=0 &\text{on } \partial\Omega, \end{cases} \end{equation*} when $\varepsilon$ tends to $0$. Here, $\Omega$ denotes a bounded open set of $\mathbb{R}^N$ $(N\geq 2)$, ${-\Delta_p u = -\mathrm{div}(|\nabla u|^{p-2}\nabla u)}$ is the usual $p$-Laplacian operator ($1<p<\infty$) and $f(x)$ is an $L^1(\Omega)$ function. We show that this sequence converges in some sense to $u$, the entropy solution of the problem \begin{equation*} \begin{cases} -\Delta_p u = f(x) &\text{in } \Omega,\\ u=0 &\text{on } \partial\Omega. \end{cases} \end{equation*} In the semilinear case, we prove stronger results provided the weak solution of that problem exists.

Keywords

• nonlinear elliptic equations
• entropy solution
• $q$-condition