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Nonlinear elliptic differential inclusions governed by state-dependent subdifferentials. (English) Zbl 0931.35203

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with a Lipschitz boundary \(\partial\Omega\) and let \(A\) be the nonlinear elliptic differential operator defined by \(Au=-\sum ^n_{i,j=1} (\partial/ \partial x_i)(a_{i,j} (x,u) \partial u/\partial x_j)\), where \(a_{i,j}: \Omega \times \mathbb{R}\to \mathbb{R}\) are Carathéodory functions satisfying some general growth conditions. Let \(f,g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) be two functions that are continuous in the first and nondecreasing in the second argument. Set, for every \((r,s)\in\mathbb{R}\times\mathbb{R}\), \(f(r,s\pm 0)= \lim _{\varepsilon \downarrow 0}f(r,s \pm\varepsilon)\) as well as \(\beta(r,s)= [f( r,s-0), f(r,s+0)]\). In this nice paper, the existence of weak solutions to the multivalued elliptic boundary value problem \(Au+\beta (u,u)\ni g(u,u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) is established under the assumption that appropriately defined upper and lower solutions are available. Moreover, the solution set enclosed by them is proved to possess extremal elements in the sense of the underlying partial ordering. The obtained result improves and unifies several previous theorems on related topics.
Reviewer: A.Marano

MSC:

35R70 PDEs with multivalued right-hand sides
35J65 Nonlinear boundary value problems for linear elliptic equations
47F05 General theory of partial differential operators
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References:

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