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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.

35J60 Nonlinear elliptic equations
47H11 Degree theory for nonlinear operators
47J05 Equations involving nonlinear operators (general)
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[1] Krasnoselskii, M. A.; Zabreiko, P. P., Geometrical methods of nonlinear analysis, (1984), Springer-Verlag Berlin, New York
[2] Volpert, A. I.; Volpert, V. A., Construction of the rotation of the vector field for operators describing wave solutions of parabolic systems, Soviet Math. Dokl., Vol. 36, n^o 3, 452-455, (1988) · Zbl 0704.35067
[3] Volpert, A. I.; Volpert, V. A., Application of the rotation theory of vector fields to the study of wave solutions of parabolic equations, Trans. Moscow Math. Soc., Vol. 52, 59-108, (1990) · Zbl 0711.35064
[4] Skrypnik, I. V., Nonlinear elliptic equations of higher order, naukova dumka, kiev, 1973 (in Russian). nonlinear elliptic boundary value problems, Teubner-Texte zur Mathematik, Vol. 91, 232, (1986), BSB Β.G. Teubner Verlagsgesellschaft Leipzig
[5] V. A. Volpert and A. I. Volpert, Travelling Waves Described by Monotone Parabolic Systems, Preprint n^o 146, CNRS URA 740, 1993, 46 p. · Zbl 0994.35049
[6] Berestycki, H.; Nirenberg, L., Travelling fronts in cylinder, Annales de l’IHP. Analyse non linéaire, Vol. 9, n^o 5, 497-572, (1992) · Zbl 0799.35073
[7] Gardner, R., Existence of multidimensional travelling wave solutions of an initial-boundary value problem, J. Diff. Eqns, Vol. 61, 335-379, (1986) · Zbl 0549.35066
[8] S. Heinze, Travelling Waves for Semilinear Parabolic Partial Differential Equations in Cyclindrical domains, Preprint n^o 506, Heidelberg, 1989, 46 p.
[9] Ladyzhenskaya, O. A.; Uraltseva, N. N., Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[10] Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear Analysis TMA, Vol. 11, n^o 10, 1103-1133, (1987) · Zbl 0639.35038
[11] Henry, D., Geometrical theory of semilinear parabolic equations, Lecture Notes in Mathematics, Vol. 840, (1981), Springer-Verlag Berlin, New York
[12] Volpert, A. I.; Hudjaev, S. I., Analysis in classes of discontinuous functions and equations of mathematical physics, (1985), Martinus Nijhoff Publisher
[13] Browder, F. E., Degree theory for nonlinear mappings, Proceedings of Symposia in Pure Mathematics, Vol. 45, Part 1, 203-226, (1986)
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