Extremal Solutions for Quasilinear Elliptic Inclusions in All of ℝN with State-Dependent Subdifferentials

S. Carl
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S. Carl: Martin-Luther-Universität Halle-Wittenberg

Journal of Optimization Theory and Applications, 2000, vol. 104, issue 2, No 4, 323-342

Abstract: Abstract This paper deals with quasilinear elliptic differential inclusions defined in all of ℝN and governed in general by a nonpotential quasilinear elliptic operator of the Leray–Lions type and a multivalued term in form of a (nonmonotone) state-dependent subdifferential. We prove the existence of entire extremal solutions within a sector of an ordered pair of appropriately defined upper and lower solutions without imposing any condition at infinity. Therefore, standard variational methods cannot be applied here. Furthermore, due to the unboundedness of the domain and due to lack of monotonicity of the operators involved, no comparison results are available such that the problem under consideration becomes even more difficult.

Keywords: Leray–Lions operators; state-dependent subdifferentials; entire extremal solutions; upper and lower solutions; unbounded domains; locally convex spaces; Fréchet spaces; gradient estimates; truncation techniques; pseudomonotone operators; eigenvalue problems (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1023/A:1004609713176

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