Zeitschrift für Analysis und ihre Anwendungen Vol. 18, No. 3, pp. 585610 (1999) 

Existence Theorems for Boundary Value Problems for Strongly Nonlinear Elliptic SystemsHô\`ng Thái Nguyê\ nSzczecin Univ., Inst. Math., ul. Wielkopolska 15, 70451 Szczecin, Poland, nguyenht@sus.univ.szczecin.plAbstract: 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 vectorvalued 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 functionaltheoretic 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 nonlinearity $F$, i.e. in the case of lack of compactness of the operator $LF$. 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 nonlinearities. Concerning solutions, we consider existence in the classical weak sense, in the socalled $L_\infty$weakened sense in both Sobolev and SobolevOrlicz spaces, and in a generalized weak sense in Sobolevtype 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. Keywords: strongly nonlinear boundary value problems, existence theorems, solution in the usual weak sense, solution in the $L_\infty$weakened sense, elliptic operators, pseudomonotone operators, generalized monotone operators, complementary systems, Sobolev spaces, SobolevOrlicz spaces, exact embedding theorems Classification (MSC2000): 35J65, 46E30, 47H10, 47H17, 47H30 Full text of the article:
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