Regularizing effect in some Mingione’s double phase problems with very singular data

Abstract

<abstract><p>In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} { } -{{{\rm{\;div}}}}\big(a(x)\,\nabla u|\nabla u|^{p-2}\big) -{{{\rm{\;div}}}}\big( |u|^{(r-1)\lambda+1}\nabla u|\nabla u|^{\lambda-2}\big) = f \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ 1 &lt; \lambda \leq p $, $ r &gt; 1 $ and $ f \in L^1(\Omega) $.</p></abstract>