Existence theorems for equations involving the 1-Laplacian: first eigenvalues for \(-\Delta_1\). (Théorèmes d’existence pour des équations avec l’opérateur ”1-Laplacien”, première valeur propre pour \(-\Delta_1\).) (French. Abridged English version) Zbl 1142.35408

The present paper is devoted to the partial differential equations of the form \[ \begin{gathered} -\text{div\,}\sigma= f(x,u),\;u\geq 0,\;u\not\equiv 0,\;u\in\text{BV}(\Omega),\\ \sigma\cdot\nabla u= |\nabla u|\quad\text{in }\Omega,\;|\sigma|_{L^\infty(\Omega)}\leq 1,\\ \sigma\cdot\vec n(-u)= u\quad\text{on }\partial\Omega,\end{gathered} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and for \(u\in \text{BV}(\Omega)\) we have \(f(x,u)\in L^N(\Omega)\). The author presents some necessary and sufficient conditions on \(f\) and the first eigenfunctions for \((-\text{div}(\sigma(u)))\) which provide existence of nontrivial solutions to \[ -\text{div}(\sigma(u))= \lambda+ fu^{q-1} \] (with boundary conditions), where \(f\in L^\infty(\Omega)\) and \(1< q\leq 1^*={N\over N-1}\).


35J60 Nonlinear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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