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Varying domains in a general class of sublinear elliptic problems. (English) Zbl 1109.35352

In this paper we use the linear theory developed in our papers [J. Differ. Equations 178, No. 1, 123–211 (2002; Zbl 1086.35073) and Nonlinear Anal., Theory Methods Appl. 47, No. 3, 1797–1808 (2001; Zbl 1042.35600)] to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the underlying domain. Our main theorem completes the results of E. N. Dancer and D. Daners [J. Differ. Equations 138, No. 1, 86–132 (1997; Zbl 0886.35063)] – and the references there in –, where the classical Robin problem was dealt with. Besides the fact that we are working with mixed non-classical boundary conditions, it must be mentioned that this paper is considering problems where bifurcation from infinity occurs; now a days, analyzing these general problems, where the coefficients are allowed to vary and eventually vanishing or changing sign, is focusing a great deal of attention – as they give rise to metasolutions (e.g.,the second author, Electron. J. Differ. Equ. 2000, Conf. 05, 135–171 (2000; Zbl 1055.35049)]).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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