On variational problems and nonlinear elliptic equations with nonstandard growth conditions

We study elliptic problems where the boundedness condition is "separated" from the coercivity condition, which leads to the loss of uniqueness, regularity, and some other properties of solutions. We propose new methods allowing us to establish the existence results for such problems, in particular, in situations where a weak solution to the Dirichlet problem is not unique and the energy equality fails. We develop a special techniques of the weak convergence of fluxes to a flux owing to which it is possible to pass to the limit in nonlinear terms. Based on this technique, we establish the solvability of the well-known thermistor problem without any restrictions on the spatial dimension and smallness of the data. Various model examples and counterexamples are also given. Bibliography: 72 titles. Illustrations: 3 figures. © 2011 Springer Science+Business Media, Inc.