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Construction of the Leray-Schauder degree for elliptic operators in unbounded domains. (English) Zbl 0835.35048
Summary: This paper is devoted to a construction of the Leray-Schauder degree for quasilinear elliptic operators in unbounded domains. The main problem here is that such operators cannot be reduced to compact ones and the usual theory cannot be applied. In previous papers of the authors, the Leray-Schauder degree was constructed in the one-dimensional case. To do this, certain lower estimates of the operators were obtained and the approach of Skrypnik was applied. In this paper, we generalize these results to the multidimensional case. When the degree is defined, the Leray-Schauder method can be used to prove the existence of solutions.

MSC:
35J60 Nonlinear elliptic equations
47H11 Degree theory for nonlinear operators
47J05 Equations involving nonlinear operators (general)
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