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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

MSC:
 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
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References:
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