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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 \(u\in K=\{v\in W_0^{1,p} (\Omega): v(x)\geq \varphi(x)\) a.e. in \(\Omega\}\) such that \[ \int_\Omega |Du|^{p-2} Du\cdot D(v-u) dx\geq \lambda \int_\Omega u^{p^*-1} (v-u) dx \qquad \forall v\in K, \tag{0.1} \] where \(\Omega\) is a bounded domain, \(2\leq p< n\), \(p^*\) a critical exponent and \(\varphi\in C^{1, \beta} (\Omega)\) \((\varphi|_{\partial \Omega}< 0\), \(\varphi^+\neq 0)\). The author has shown if \(\lambda\) 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.

MSC:

49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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