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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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[1] Alama, S.; Tarantello, G., On semilinear elliptic equations with indefinite nonlinearities, Calc. var. and partial differential equations, 1, 439-475, (1993) · Zbl 0809.35022
[2] Berestycki, H.; Capuzzo Dolcetta, I.; Nirenberg, L., Problémes elliptiques indéfinis et théorème de Liouville non-linéaires, C. R. acad. sci. Paris, Série I, 317, 945-950, (1993) · Zbl 0820.35056
[3] I. Birindelli, F. Demengel, On some partial differential equation for noncoercive functional and critical Sobolev exponent, Adv. in Differential Equations, accepted · Zbl 1045.35034
[4] I. Birindelli, F. Demengel, On some partial differential equation for non coercive functional and critical Sobolev exponent, Preprint, Universita di Roma La Sapienza · Zbl 1045.35034
[5] Demengel, F., On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent, Control optim. calc. var., (Mars 2000)
[6] F. Demengel, Some existence’s results for noncoercive 1-Laplacian operator, Prébublication de l’Université de Cergy-Pontoise, No. 21, 2001, soumis à Nonlinear Anal
[7] F. Demengel, Functions almost 1-harmonic, Prépublication de l’Université de Cergy Pontoise. No. 31, 2001
[8] Lions, P.-L., La méthode de compacité concentration, I et II, Rev. mat. iberoamericana, 1, 1, 145, (1985)
[9] Ouyang, T., On the positive solutions of semilinear equations of δu+λu+hup=0 on compacts manifolds, II, Indiana math. J., 40, 1083-1140, (1991)
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