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Remarks on perturbated systems with critical growth. (English) Zbl 0786.35061
The author considers the question on convergence almost everywhere for gradients \(Du_ j\) of a bounded sequence \(u_ j\) in the Sobolev space \(W_ 0^{1,p} (\Omega;\mathbb{R}^ N)\) which appears in the study of the Dirichlet problem for the elliptic system with critical growth perturbation: \[ A(u)+G(u)=f,\quad u |_{\partial \Omega} =0 \] where \(\Omega \subset \mathbb{R}^ n\) is a bounded domain, \(A(u)\) is a quasilinear operator and the perturbation \(G(u)\) is dependent on the (vector-valued) function \(u\) and its gradient \(Du\). The growth of the perturbation is called critical if it can be controlled by \(| Du |^ p\) only, where \(p\) is the same exponent for which \(A(u)\) defines a coercive mapping from \(W_ 0^{1,p}(\Omega;\mathbb{R}^ N)\) to its dual space \((W_ 0^{1,p}(\Omega;\mathbb{R}^ N))^*\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Archs ration. mech. analysis, 86, 125-145, (1984) · Zbl 0565.49010
[2] Adams, R.M., Sobolev spaces, (1975), Academic Press New York
[3] Balder, E.J., A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. control optim., 22, 570-597, (1984) · Zbl 0549.49005
[4] Balder E. J., On infinite-horizon lower closure results for optimal control, Annali Mat. pura appl. (to appear). · Zbl 0664.49005
[5] Ball, J.M., A version of the fundamental theorem of Young measures, (), (to appear). · Zbl 0991.49500
[6] Ball, J.M.; Murat, F., Remarks on Chacon’s biting lemma, ESCP mathl probl. nonlinear mech. preprint ser., 14, (1988) · Zbl 0678.46023
[7] Ball, J.M.; Zhang, K., Lower semicontinuity of multiple integrals and the biting lemma, Proc. R. soc. edinb., 114A, 367-379, (1990) · Zbl 0716.49011
[8] Berliocchi, H.; Lasry, J.M., Integrandes normales et mesures parametres en calcul des variations, Bull. soc. math. fr., 101, 129-184, (1973) · Zbl 0282.49041
[9] Brooks, J.K.; Chacon, R.V., Continuity and compactness of measures, Adv. math., 37, 16-26, (1980) · Zbl 0463.28003
[10] Browder, F.E., Theorems for nonlinear partial differential equations, Proc. symp. pure math., 16, 1-60, (1970) · Zbl 0212.27704
[11] Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, () · Zbl 0676.46035
[12] Eisen, G., A selection lemma for sequences of measurable sets and lower semicontinuity of multiple integrals, Manuscripta math., 27, 73-79, (1979) · Zbl 0404.28004
[13] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationnels, (1974), Dunod Paris · Zbl 0281.49001
[14] Frehse, J., Existence and perturbation theorems for nonlinear elliptic systems, Preprint no. 576, (1983), SFB 72, Bonn · Zbl 0528.35036
[15] Landes, R., On the existence of weak solutions of perturbated systems with critical growth, J. reine angew. math., 393, 21-38, (1989) · Zbl 0664.35027
[16] Landes, R.; Mustonen, V., On pseudomonotone operators and nonlinear noncoercive variational problems on unbounded domains, Math. ann., 248, 241-246, (1980) · Zbl 0416.35072
[17] Neumann, J.von, Functional operators: measures and integrals, Volume I, (1950), Princeton University Press Princeton, New Jersey
[18] Schonbek, M.E., Convergence of solutions to nonlinear dispersive equations, Communs partial diff. eqns, 7, 959-1000, (1982) · Zbl 0496.35058
[19] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, New Jersey · Zbl 0207.13501
[20] Tartar, L.; Knops, R.J., Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: heriot-watt symposium, Vol. IV, 136-212, (1979)
[21] Zhang, K.-W., On the Dirichlet problem for a class of quasilinear elliptic systems of PDEs in divergence form, (), 262-277, Tianjin, 1986
[22] Zhang, K.-W., Nash point equilibria for variational integrals, I—existence results, Acta math. sinica, new ser., 4, 155-176, (1988) · Zbl 0850.49002
[23] Zhang, K.-W., Biting theorems for Jacobians and their applications, Anal. non lin. ann. H. Poincaré, 7, 345-365, (1990) · Zbl 0717.49012
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