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Strongly nonlinear unilateral problems. (English) Zbl 0535.49010
The authors prove the existence of a solution for the obstacle problem of the type \[ u\in K,\quad g(.,u)\in L^ 1(\Omega),\quad g(.,u)u\in L^ 1(\Omega), \] \[ <A(u)-f,v-u>+\int_{\Omega}g(.,u)(v-u)\geq 0\quad for\quad all\quad v\in K\cap L^{\infty}(\Omega), \] where A is a nonlinear differential operator of order 2m (\(m\geq 1)\) satisfying the classical Leray-Lions conditions and giving rise to a ”good” operator on \(H^{m,p}(\Omega) (1<p<\infty)\), g is a strongly nonlinear term satisfying essentially the sign condition, V is either \(H^{m,p}(\Omega)\) or \(H_ 0\!^{m,p}(\Omega)\), \(f\in V^*\) and \(K=\{v\in V : v\geq \psi\) in \(\Omega\}\) with \(\psi \in V\cap L^{\infty}(\Omega)\) and \(\Omega\) some domain in \({\mathbb{R}}^ n.\)
An essential tool in the proof is the approximation result by L. J. Hedberg [Acta Math. 147, 237-264 (1981; Zbl 0504.35018)]. Some further properties of the solution are also proved and the bilateral obstacle problem is discussed. It is not hard to see that the restrictive assumption \(\psi\in V\) can be removed.
Reviewer: V.Mustonen

49J40 Variational inequalities
35J60 Nonlinear elliptic equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
47J05 Equations involving nonlinear operators (general)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
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