JournalszaaVol. 18, No. 3pp. 585–610

Existence Theorems for Boundary Value Problems for Strongly Nonlinear Elliptic Systems

  • Hong Thai Nguyen

    Szczecin University, Poland
Existence Theorems for Boundary Value Problems for Strongly Nonlinear Elliptic Systems cover
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Abstract

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

DOI 10.4171/ZAA/900