# zbMATH — the first resource for mathematics

On the existence of weak solutions for a class of elliptic partial differential systems. (English) Zbl 1219.35088
Summary: We obtain the existence of weak solutions for $\sum_{\alpha =1}^{n}\frac{\partial }{\partial x^{\alpha }} A_{\alpha }^{i}\left( x,u,Du\right) =B^{i}\left( x,u,Du\right),~ x\in\Omega,~ i=1,\dots ,N.$ in a Orlicz-Sobolev space. This generalizes the result of E. Acerbi and N. Fusco [Arch. Ration. Mech. Anal. 86, 125–145 (1984; Zbl 0565.49010)] by means of Orlicz space theory.

##### MSC:
 35J56 Boundary value problems for first-order elliptic systems 35J50 Variational methods for elliptic systems 35J60 Nonlinear elliptic equations
Full Text:
##### References:
 [1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. rational mech. anal., 86, 125-135, (1984) · Zbl 0565.49010 [2] Adams, R.A., Sobolev space, (1975), Academic Press New York [3] Benkirane, A.; Elmahi, A., An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear anal., 36, 11-24, (1999) · Zbl 0920.35057 [4] Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture notes in mathematics, Vol. 922, (1982), Springer Berlin [5] Donaldson, T.K., Nonlinear elliptic boundary value problems in orlicz – sobolev space, J. differential equation, 10, 507-528, (1971) · Zbl 0207.41501 [6] Donaldson, T.K.; Trudinger, N.S., Orlicz-Sobolev spaces and imbedding theorems, J. funct. anal., 8, 52-75, (1971) · Zbl 0216.15702 [7] Eisen, G., A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals, Manuscript math., 27, 73-79, (1979) · Zbl 0404.28004 [8] Ekeland, I.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam [9] Gossez, J.P., Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. amer. math. soc., 190, 163-205, (1974) · Zbl 0239.35045 [10] Gossez, J.P., A strongly nonlinear elliptic problem in orlicz – sobolev spaces, Proc. AMS symp. pure math., 45, 455-462, (1986) [11] M.A. Krasnoselski, Ya.B. Rutickii, Convex functions and Orlicz space, Noordhoff, Groningen, 1961. [12] Lieberman, G.M., The natural generalization of the natural conditions of ladyzhenskaya and ural’tseva for elliptic equations, Comm. partial differential equations, 16, 311-361, (1991) · Zbl 0742.35028 [13] Liu, F.C., A luzin type property of Sobolev functions, Indiana univ. math. J., 26, 645-651, (1977) · Zbl 0368.46036 [14] Morrey, C.B., Multiple integrals in the calculus of variations, (1966), Springer New York · Zbl 0142.38701 [15] Visik, M.I., Solvability of the first boundary value problem for quasilinear equations with rapidly increasing coefficients in Orlicz classes, Dokl. akad. nauk. SSSR, 151, 758-761, (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.