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Semilinear elliptic equations in unbounded domains of Rn

  • Patrizia Donato (a1), Lucia Migliaccio (a2) and Rosanna Schianchi (a2)


We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.



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1Adams, R. A.. Sobolev spaces (London: Academic Press, 1971).
2Agmon, S.. The Lp approach to the Dirichlet problem I. Ann. Scuola Norm. Sup. Pisa 13 (1959), 405448.
3Agmon, S.. On the eigenfunction and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl Math. 15 (1962), 119147.
4Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18 (1976), 620709.
5Amann, H.. Existence and multiplicity theorems for semilinear elliptic value problems. Math. Z. 150 (1976) 281295.
6Amann, H.. Nonlinear operators in ordered Banach spaces and some appications to nonlinear boundary value problems. In Nonlinear operators and the calculus of the variations. Lecture Notes in Mathematics 573 (Berlin: Springer, 1976).
7Amann, H. and Crandall, M. G.. On the existence theorems for semilinear elliptic equations. Indiana Univ. Math. J. 27 (1978), 779790.
8Benci, V. and Fortunate, D.Some compact embedding theorems for weighted Sobolev spaces. Boll. Un. Mat. Ital. 13B (1976), 832843.
9Benci, V. and Fortunato, D.. Weighted Sobolev spaces and nonlinear Dirichlet problem in unbounded domains. Ann. Mat. Pura Appl., in press.
10Bony, J. M.. Principles de maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris, Ser. A 265 (1976), 333336.
11Matarasso, S. and Troisi, M.. Operatori differenziali ellittici in spazi di Sobolev con peso. Ricerche Mat. 27 (1978) 1, 88108.
12Protter, M. H. and Weinberger, H. F.. Maximum principles in differential equations (Englewood Cliffs, N.J.; Prentice Hall, 1967).
13Tomi, F.. Über semilineare elliptische Differentialgleichungen zweiter ordnung. Math. Z. 111 (1969), 350366.
14Troisi, M.. Teoremi di inclusione negli spazi di Sobolev con peso. Ricerche Mat. 18 (1969), 4974.
15Wahl, W. V.. Über quasilineare ellptische Differentialgleichungen in der Ebene. Manuscripta Math. 8 (1973), 5967.


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