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Existence of bounded solutions for non linear elliptic unilateral problems. (English) Zbl 0687.35042
Let Ω be a bounded domain in n , ψ a measurable function on Ω, p>1, and A(u)=diva(x,u,Du)+a 0 (x,u,Du) an elliptic quasilinear differential operator whose coefficients a, a 0 satisfy natural regularity and growth conditions which in particular guarantee that A is a continuous, pseudomonotone operator from the Sobolev space W 0 1,p (Ω) into its dual. The authors prove the existence of a solution uW 0 1,p (Ω)L (Ω) of the variational inequality uψ, <A(u),v-u>+H(x,u,Du)(v-u)dx0 for all vW 0 1,p (Ω)L (Ω) such that vψ. Here it is important to notice that the inhomogeneous term H is allowed to grow like |Du| p . The proof is carried out by an approximation of H by bounded functions H ϵ for which a solution u ϵ of the corresponding problem is known to exist. Then it is shown that the family u ϵ is compact in W 0 1,p (Ω).
Reviewer: F.Tomi

MSC:
35J85Unilateral problems; variational inequalities (elliptic type) (MSC2000)
47H05Monotone operators (with respect to duality) and generalizations
35J65Nonlinear boundary value problems for linear elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
49J40Variational methods including variational inequalities
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