In this paper we study the Dirichlet problem for two related equations involving the 1-Laplacian and a total variation term as reaction, namely:

with homogeneous Dirichlet boundary conditions on \(\partial \varOmega \), where \(\varOmega \) is a regular, bounded domain in \(\mathbb {R}^N\). Here f is a measurable function belonging to some suitable Lebesgue space, while g(u) is a continuous function having the same sign as u and such that \(g(\pm \infty ) = \pm \infty \). As far as Eq. (1) is concerned, we show that a bounded solution exists if the datum f belongs to \(L^N(\varOmega )\). When the absorption term g(u) is missing, i.e. in the case of Eq. (2), we show that if \(f\in L^N(\varOmega )\), and its norm is small, then the only solution of (2) is \(u\equiv 0\). In the case where the norm of f is not small, several cases may happen. Depending on \(\varOmega \) and f, we show examples where no solution of (2) exists, other examples where \(u\equiv 0\) is still a solution, and finally examples with nontrivial solutions. Some of these results can be viewed as a translation to the 1-Laplacian operator of known results by Ferone and Murat.

中文翻译：

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具有总变化项的1-Laplacian方程的有界解

在本文中，我们研究了涉及1-Laplacian和总变化项作为反应的两个相关方程的Dirichlet问题，即：

在\（\ partial \ varOmega \）上具有齐次Dirichlet边界条件，其中\（\ varOmega \）是\（\ mathbb {R} ^ N \）中的规则有界域。这里f是属于某个合适的Lebesgue空间的可测量函数，而g（u）是具有与u相同符号并且使得\（g（\ pm \ infty）= \ pm \ infty \）的连续函数。据等式。关于（1），我们证明如果基准面f属于\（L ^ N（\ varOmega）\），则存在有界解。当吸收项g（u）丢失，即在等式的情况下。（2），我们证明如果\（f \ in L ^ N（\ varOmega）\），并且其范数很小，则（2）的唯一解是\（u \ equiv 0 \）。在f的范数不小的情况下，可能会发生几种情况。根据\（\ varOmega \）和f，我们显示不存在（2）的解的示例，还显示\（u \ equiv 0 \）仍然是解的示例，最后给出具有平凡解的示例。其中一些结果可以看作是Ferone和Murat将已知结果的1-Laplacian运算符的翻译。