Existence Theorems for Boundary Value Problems for Strongly Nonlinear Elliptic Systems

Abstract

Let $L$ be a linear elliptic, a pseudomonotone or a generalized monotone operator (in the sense of F. E. Browder and I. V. Skrypnik), and let $F$ be the nonlinear Nemytskij superposition operator generated by a vector-valued function $f$. We give two general existence theorems for solutions of boundary value problems for the equation $Lx=Fx$. These theorems are based on a new functional-theoretic approach to the pair $(L,F)$, on the one hand, and on recent results on the operator $F$, on the other hand. We treat the above mentioned problems in the case of strong non-linearity $F$, i.e. in the case of lack of compactness of the operator $L-F$. In particular, we do not impose the usual growth conditions on the nonlinear function $f$; 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 $L_{\infty }$-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 $L_{\infty }$-modules. Finally, we illustrate the abstract results by some applied problems occuring in nonlinear mechanics.