Abstract
We consider the problem of passing to the limit in a sequence of nonlinear elliptic problems. The “limit” equation is known in advance, but it has a nonclassical structure; namely, it contains the p-Laplacian with variable exponent p = p(x). Such equations typically exhibit a special kind of nonuniqueness, known as the Lavrent’ev effect, and this is what makes passing to the limit nontrivial. Equations involving the p(x)-Laplacian occur in many problems of mathematical physics. Some applications are included in the present paper. In particular, we suggest an approach to the solvability analysis of a well-known coupled system in non-Newtonian hydrodynamics (“stationary thermo-rheological viscous flows”) without resorting to any smallness conditions.
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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 43, No. 2, pp. 19–38, 2009
Original Russian Text Copyright © by V. V. Zhikov
Supported by the Russian Foundation for Basic Research under grant no. 08-01-99007.
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Zhikov, V.V. On the technique for passing to the limit in nonlinear elliptic equations. Funct Anal Its Appl 43, 96–112 (2009). https://doi.org/10.1007/s10688-009-0014-1
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DOI: https://doi.org/10.1007/s10688-009-0014-1