Boccardo, Lucio; Giachetti, Daniela Strongly nonlinear unilateral problems. (English) Zbl 0535.49010 Appl. Math. Optimization 9, 291-301 (1983). 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 Cited in 1 ReviewCited in 9 Documents 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 Keywords:nonlinear unilateral problems; obstacle problem Citations:Zbl 0504.35018 PDFBibTeX XMLCite \textit{L. Boccardo} and \textit{D. Giachetti}, Appl. Math. Optim. 9, 291--301 (1983; Zbl 0535.49010) Full Text: DOI References: [1] Br?zis H (1968) Equations et in?quations non lin?aires dans les espaces vectoriels en dualit?. Ann Inst Fourier (Grenoble) 18:115-175 · Zbl 0169.18602 [2] Br?zis H, Browder FE (1978) Strongly nonlinear elliptic boundary value problems. Ann Sc Norm Sup Pisa 5:587-603 · Zbl 0453.35029 [3] Br?zis H, Broder FE (to appear) Some properties of higher order Sobolev spaces. J Math Pures Appl [4] Browder FE (1977) Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc Nat Acad Sci 74:2659-2661 · Zbl 0358.35034 · doi:10.1073/pnas.74.7.2659 [5] Edmunds DE, Webb JRL (1973) Quasilinear elliptic problems in unbounded domains. Proc Roy Soc London, Ser A, 337:397-410 · Zbl 0263.35034 [6] Hedberg LI (1978) Two approximation problems in function spaces. Ark Mat 16:51-81 · Zbl 0399.46023 · doi:10.1007/BF02385982 [7] Hedberg LI (to appear) Spectral synthesis in Sobolev spaces. Acta Math [8] Hess P (1973) Variational inequalities for strongly nonlinear elliptic operators. J Math Pures Appl 52:285-298 · Zbl 0222.47020 [9] Hess P (1974) On a class of strongly nonlinear elliptic variational inequalities. Math Ann 211:289-297 · Zbl 0285.49003 · doi:10.1007/BF01418226 [10] Mustonen V (1978) A class of strongly nonlinear variational inequalities. J London Math Soc (2) 18:157-165 · Zbl 0392.35022 · doi:10.1112/jlms/s2-18.1.157 [11] Mustonen V (1979) A class of strongly nonlinear variational inequalities in unbounded domains. J. London Math Soc (2) 19:319-328 · Zbl 0404.35035 · doi:10.1112/jlms/s2-19.2.319 [12] Webb JRL (1980) Boundary value problems for strongly nonlinear elliptic equations. J London Math Soc (2) 21:123-132 · Zbl 0438.35029 · doi:10.1112/jlms/s2-21.1.123 [13] Stein EM (1970) Singular integrals and differentiability properties of functions. Princeton University Press, Princeton · Zbl 0207.13501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.