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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 $\varphi\in C^{1, \beta} (\Omega)$ $(\varphi|_{\partial \Omega}< 0$, $\varphi^+\ne 0)$. 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.

49J40Variational methods including variational inequalities
35J85Unilateral problems; variational inequalities (elliptic type) (MSC2000)
Full Text: DOI
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