Calculus of Variations and Geometric Measure Theory
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D. Giachetti - F. Oliva - F. Petitta

1-Laplacian type problems with strongly singular nonlinearities and gradient terms

created by petitta on 30 Oct 2019
modified on 23 Sep 2021


Accepted Paper

Inserted: 30 oct 2019
Last Updated: 23 sep 2021

Journal: Comm. Contemp. Math
Year: 2021

ArXiv: 1910.13311 PDF


We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -\Delta1 u = g(u)
D u
+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(\Omega)$, and $g$ and $h$ are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities $g$ and $h$ produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.

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