Let be a linear elliptic, a pseudomonotone or a generalized monotone operator (in the sense of F. E. Browder and I. V. Skrypnik), and let be the nonlinear Nemytskij superposition operator generated by a vector-valued function . We give two general existence theorems for solutions of boundary value problems for the equation . These theorems are based on a new functional-theoretic approach to the pair , on the one hand, and on recent results on the operator , on the other hand. We treat the above mentioned problems in the case of strong non-linearity , i.e. in the case of lack of compactness of the operator . In particular, we do not impose the usual growth conditions on the nonlinear function ; this allows us to treat elliptic systems with rapidly growing coefficients or exponential non-linearities. Concerning solutions, we consider existence in the classical weak sense, in the so-called -weakened sense in both Sobolev and Sobolev-Orlicz spaces, and in a generalized weak sense in Sobolev-type spaces which are modelled by means of Banach -modules. Finally, we illustrate the abstract results by some applied problems occuring in nonlinear mechanics.
Cite this article
Hong Thai Nguyen, Existence Theorems for Boundary Value Problems for Strongly Nonlinear Elliptic Systems. Z. Anal. Anwend. 18 (1999), no. 3, pp. 585–610