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arXiv · 1910.13311

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

Daniela Giachetti·Francescantonio Oliva·Francesco Petitta
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Abstract

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -\Delta_1 u = g(u)|D u|+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} $$ 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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Daniela Giachetti, Francescantonio Oliva, Francesco Petitta. 2019-10-29. 1-Laplacian type problems with strongly singular nonlinearities and gradient terms. https://doi.org/10.1142/s0219199721500814

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