# Existence results for monotone perturbations of the Dirichlet problem.

Chipot, M; Kis, L (2000). Existence results for monotone perturbations of the Dirichlet problem. Advances in Mathematical Sciences and Applications, 10(1):81-101.

## Abstract

Let Ω⊂ℝ n , n≥1, be a bounded domain and A an elliptic operator given by Au=∑ i,j=1 n ∂ ∂x i (a ij (x)∂u ∂x j ), with a ij ∈L ∞ (Ω), 1≤i, j≤n, a ij (x)ξ i ξ j ≥d|ξ| 2 , ξ∈ℝ n , a.e. x∈Ω. For f∈L p (Ω) the authors study existence results for problems of the form: -Au+β(x,u)∋f in Ω, u=0 on ∂Ω, where β(x,·) a maximal monotone graph. This extends a well-known theory of H. Brezis (see [Problèmes unilatéraux, J. Math. Pure Appl. 51, 1-162 (1972); Noveaux théorèmes de regularité pur les problèmes unilatéraux, Recontres entre physiciens théoriciens et mathématiciens 12 (1971), Strasbourg]) to the case where β depends on x.

## Abstract

Let Ω⊂ℝ n , n≥1, be a bounded domain and A an elliptic operator given by Au=∑ i,j=1 n ∂ ∂x i (a ij (x)∂u ∂x j ), with a ij ∈L ∞ (Ω), 1≤i, j≤n, a ij (x)ξ i ξ j ≥d|ξ| 2 , ξ∈ℝ n , a.e. x∈Ω. For f∈L p (Ω) the authors study existence results for problems of the form: -Au+β(x,u)∋f in Ω, u=0 on ∂Ω, where β(x,·) a maximal monotone graph. This extends a well-known theory of H. Brezis (see [Problèmes unilatéraux, J. Math. Pure Appl. 51, 1-162 (1972); Noveaux théorèmes de regularité pur les problèmes unilatéraux, Recontres entre physiciens théoriciens et mathématiciens 12 (1971), Strasbourg]) to the case where β depends on x.
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