ABSTRACT
Throughout, we shall use the method of nonlinear capacity suggested by Pohozaev in 1997 [328] and further developed jointly with Mitidieri in [302, 303]. Actually, earlier, in some particular and more standard cases (e.g., for some semilinear parabolic and hyperbolic equations), we have used such an approach. The essence of this method consists of the reduction of the original problem
(independent of the type of the equation) with a related variational one. The extreme value of the respective functional generates the nonlinear capacity associated to the original problem. The nonlinear capacity equals the optimal a priori estimate in the respective class of test functions. Its behavior determines the existence or nonexistence of a global solution. We note that, in the applications, it is not necessary to find the exact
extremal value. In order to obtain sufficient conditions of blow-up, it is enough to use nearly optimal test functions. This approach allows us to establish a homotopic invariance of the criti-
cal exponents. In [303], a “Mendeleev-type Table of Elements” of nonlinear operators, together with their critical exponents, is presented.
