Geodesic
On the Technique for Passing to the Limit in Nonlinear Elliptic Equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 19-38

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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.
Keywords: $p(x)$-Laplacian, compensated compactness, weak convergence of flows to a flow. 
@article{FAA_2009_43_2_a1,
     author = {V. V. Zhikov},
     title = {On the {Technique} for {Passing} to the {Limit} in {Nonlinear} {Elliptic} {Equations}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {19--38},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a1/}
}
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V. V. Zhikov. On the Technique for Passing to the Limit in Nonlinear Elliptic Equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 19-38. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a1/