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Positive solutions of quasilinear elliptic obstacle problems with critical exponents. (English) Zbl 0838.49008

In this paper it has been considered a problem of finding positive solutions for a quasilinear elliptic obstacle problem with a critical exponent: find uK={vW 0 1,p (Ω):v(x)ϕ(x) a.e. in Ω} such that

Ω |Du| p-2 Du·D(v-u)dxλ Ω u p * -1 (v-u)dxvK,(0·1)

where Ω is a bounded domain, 2p<n, p * a critical exponent and ϕC 1,β (Ω) (ϕ| Ω <0, ϕ + 0). The author has shown if λ is not too big that (0.1) has a minimal positive solution by using the Ekeland’s variational principle and that in some cases the problem (0.1) has at least two positive solutions by using a variant mountain pass theorem.

49J40Variational methods including variational inequalities
35J85Unilateral problems; variational inequalities (elliptic type) (MSC2000)