Regularizing effect of absorption terms in singular problems
Abstract
We prove existence of solutions to problems whose model is $$\begin{cases} \displaystyle \Delta_p u + u^q = \frac{f}{u^\gamma} & \text{in}\ \Omega, \newline u\ge0 &\text{in}\ \Omega,\newline u=0 &\text{on}\ \partial\Omega, \end{cases}$$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $\Delta_p u$ is the $p$laplacian operator for $1\le p <N$, $q>0$, $\gamma\ge 0$ and $f$ is a nonnegative function in $L^m(\Omega)$ for some $m\ge1$. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case $\gamma\le 1$. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local $W^{1,p}$solutions in case $\gamma>1$.
 Publication:

arXiv eprints
 Pub Date:
 October 2018
 arXiv:
 arXiv:1811.00083
 Bibcode:
 2018arXiv181100083O
 Keywords:

 Mathematics  Analysis of PDEs