On the Technique for Passing to the Limit in Nonlinear Elliptic Equations

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.