## 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 $$1< q\leq 1^*={N\over N-1}$$.

### MSC:

 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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### References:

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