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On some nonlinear partial differential equations involving the 1-Laplacian

Mouna Kraïem — 2007

Annales de la faculté des sciences de Toulouse Mathématiques

Let Ω be a smooth bounded domain in N , N > 1 and let n * . We prove here the existence of nonnegative solutions u n in B V ( Ω ) , to the problem ( P n ) - div σ + 2 n Ω u - 1 sign + ( u ) = 0 in Ω , σ · u = | u | in Ω , u is not identically zero , - σ · n u = u on Ω , where n denotes the unit outer normal to Ω , and sign + ( u ) denotes some L ( Ω ) function defined as: sign + ( u ) . u = u + , 0 sign + ( u ) 1 . Moreover, we prove the tight convergence of u n towards one of the first eingenfunctions for the first 1 - Laplacian Operator - Δ 1 on Ω when n goes to + .

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