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On some nonlinear partial differential equations involving the 1-Laplacian. (English) Zbl 1184.35148

The author deals with the existence of nonnegative solutions \(w_n\) in \(BV(\Omega)\), to the problem
\[ \begin{cases}\text{-div}\,\sigma+2n (\int_\Omega w-1) \text{sign}^+(w)=0 & \text{in } \Omega,\\ \sigma \cdot\nabla w=|\nabla w| & \text{in }\Omega,\\ w\text{ is not identically zero, }-\sigma\cdot\vec nw=w &\text{in }\Omega,\end{cases} \tag{1} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\), \(N>1\), \(\vec n\) denotes the unit outer normal to \(\partial \Omega\), and \(\text{sign}^+(w)\) denotes some \(L^\infty(\Omega)\) function defined as
\[ \text{sign}^+(w)w=w^+,\quad 0\leq\text{sign}^+(w)\leq 1. \]
The purpose of the present paper is to propose an approach of the first eigenvalue and first eigenfunctions, using a penalization method.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
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