Zeitschrift für Analysis und ihre Anwendungen


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Volume 29, Issue 1, 2010, pp. 63–75
DOI: 10.4171/ZAA/1397

Published online: 2010-01-04

Entire Extremal Solutions for Elliptic Inclusions of Clarke’s Gradient Type

Patrick Winkert[1]

(1) Universität Halle-Wittenberg, Germany

We consider multivalued quasilinear elliptic problems of hemivariational type in all of ℝN given by

−∆pu + ∂j(·, u) ∋ 0 in D',
and show the existence of entire extremal solutions by applying the method of sub- and supersolutions without imposing any condition at infinity. Due to the unboundedness of the domain, standard variational methods cannot be applied. The novelty of our approach is on the one hand to obtain entire solutions and on the other hand that Clarke’s generalized gradient need only satisfies a natural growth condition. In the last section conditions are provided that ensure the existence of nontrivial positive solutions.

Keywords: Elliptic inclusions, Clarke’s generalized gradient, p-Laplacian, sub-supersolution, unbounded domains, Fréchet spaces, locally convex spaces

Winkert Patrick: Entire Extremal Solutions for Elliptic Inclusions of Clarke’s Gradient Type. Z. Anal. Anwend. 29 (2010), 63-75. doi: 10.4171/ZAA/1397