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Nonlinear elliptic equations with measure data. (English) Zbl 0815.35023

Summary: We prove the existence of solutions of nonlinear equations of the type \(- \text{div} (a(x,u,Du) + H(x,u,Du))= f\), where \(a\) and \(H\) are Caratheodory functions and \(f\) is a bounded Radon measure. We remark that the operator can be not coercive. We give also some regularity results.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data

Keywords:

measure data
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[1] Alvino, A.: Formule di maggiorazione e regolarizzazione per soluzioni di equazioni ellittiche del secondo ordine in un caso limite,Atti Acc. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 52(8) (1977), 335-340. · Zbl 0371.35009
[2] Alvino, A., Lions, P. L. and Trombetti, G.: Comparison results for elliptic and parabolic equations via Schwartz symmetrization,Ann. Inst. Henry Poincaré 7(2) (1990), 37-65. · Zbl 0703.35007
[3] Alvino, A. and Trombetti, G.: Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri,Ricerche Mat. 27 (1978), 413-428. · Zbl 0403.35027
[4] Brezis, H.:Some Variational Problems of the Thomas-Fermi Type, Variational Inequalities and Complementarity Problems, Cottle-Giannessi and Lions (eds.), J. Wiley and Sons (1980), 53-73.
[5] Brezis, H.:Nonlinear Elliptic Equations Involving Measures, Contributions to nonlinear partial differential equations, C. Bardos, A. Damlamian, J. I. Diaz and J. Hernandez (eds), Pitman Research Notes in Mathematics, Series 89 (1983), 82-89.
[6] Benilan, P., Brezis, H. and Crandall, M.: A semilinear equation inL 1,Ann. Scuola Norm. Sup. Pisa,2 (1975), 523-555. · Zbl 0314.35077
[7] Bensoussan, A., Boccardo, L. and Murat, F.: On a nonlinear partial differential equation having natural growth terms and unbounded solution,Analyse Nonlinéaire, Annales Inst. H. Poincaré,5(6) (1988), 347-364. · Zbl 0696.35042
[8] Brezis, H. and Friedman, A.: Nonlinear parabolic equations involving measures as initial conditions,J. Math. pures et appl. 62 (1983), 73-97. · Zbl 0527.35043
[9] Betta, M. F., Ferone, V. and Mercaldo, A.: Regularity for solutions of nonlinear elliptic equations, Preprint n.56 Dip. Mat. Appl. Università di Napoli (1992). To appear inBull. Sci. Math. · Zbl 0842.35014
[10] Boccardo, L. and Gallouet, T.: Nonlinear elliptic and parabolic equations involving measure data,Journal of Functional Analysis 87(1) (1989), 149-169. · Zbl 0707.35060
[11] Boccardo, L. and Gallouet, T.: Nonlinear elliptic equations with right hand side measure,Comm. in P.D.E. 17(3 e 4) (1992), 641-654. · Zbl 0812.35043
[12] Boccardo, L. and Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations,Nonlinear Analysis, Theory, M. and A. 19(6) (1992), 581-597. · Zbl 0783.35020
[13] Betta, M. F. and Mercaldo, A.: Comparison and regularity results for a nonlinear elliptic equation,Nonlinear Anal. T.M.A. 20(1) (1993), 63-77. · Zbl 0786.35008
[14] Bottaro, G.: Alcune condizioni sufficienti per l’esistenza e l’unicità della soluzione di una disequazione variazionale non coerciva,Annali di Matematica Pura e Applicata, Serie IV (1975), 187-203. · Zbl 0321.49004
[15] Baras, P. and Pierre, M.: Critère d’existence de solution positives pour des equations semi-linèaires non monotones,Ann. Inst. H. Poincaré 3 (1985), 185-212. · Zbl 0599.35073
[16] Bennett, C. and Sharpley, R.:Interpolation of Operators, Academic press, 1988. · Zbl 0647.46057
[17] Brezis, H. and Strauss, W.: Semilinear elliptic equations inL 1,J. Math. Soc. Japan 25 (1973), 565-590. · Zbl 0278.35041
[18] Brezis, H. and Veron, L.: Removable singularities for some nonlinear elliptic equations,Archive for Rational Mechanics and Analysis 75 (1980), 1-6. · Zbl 0459.35032
[19] Ferone, V.: Estimates and regularity for solutions of elliptic equations in a limit case,Boll.U.M.I. (7) Vol. 8-B (1994), 257-270. · Zbl 0799.35056
[20] Gallouet, T.: Equations elliptiques semilinéaires avec, pour la non linéarité, une condition de signe et une dépendence sous-quadratique par rapport au gradient,Ann. Fac. Sc. Univ. Toulouse 9(2) (1988), 161-169.
[21] Gallouet, T. and Morel, J. M.: Resolution of a semilinear equation inL 1,Proc. Roy. Soc. Edinburg Sect. A 96 (1984), 275-288.
[22] Gallouet, T. and Morel, J. M.: On some semilinear problems inL 1,Boll. Un. Mat. Ital. 4-A(6) (1985), 121-131.
[23] Lions, J. L.:Quelques méthodes de rèsolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. · Zbl 0189.40603
[24] Liang, J.: Weakly-coercive quasilinear elliptic equations with inhomogeneous measure data, Progress in P.D.E.: Ellip. and parab. problems ? Pitman Research Notes in Math.266 (1992).
[25] Leray, J. and Lions, J. L.: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les methodes de Minty-Browden,Bull. Soc. Math. France 93 (1965), 97-107. · Zbl 0132.10502
[26] Boccardo, L., Diaz, J. I., Giachetti, D. and Murat, F.:Existence of a Solution for a Weaker Form of a Nonlinear Elliptic Equation, Recent advances in nonlinear elliptic and parabolic problems, P. Benilan, M. Chipot, L. C. Evans, and M. Pierre (eds.), Pitman Research Notes in Mathematics, Series 208. · Zbl 0703.35063
[27] Rakotoson, J. M.: Quasilinear elliptic problems with measures as data,Differential and Integral Equations 4(3) (1991), 449-457. · Zbl 0834.35056
[28] Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus,Ann. Inst. Fourier Grenoble 15 (1965), 189-258. · Zbl 0151.15401
[29] Talenti, G.: Elliptic equations and rearrangements,Ann. Scuola Norm. Sup. Pisa 3(4) (1976), 697-718. · Zbl 0341.35031
[30] Talenti, G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces,Ann. Mat. Pura e applicata 120(4) (1979), 159-184. · Zbl 0419.35041
[31] Talenti, G.: Linear elliptic P.D.E.’s: Level sets, rearrangements and a priori estimates of solutions,Boll. Un. Mat. Ital. 4-B(6) (1985), 917-949. · Zbl 0602.35025
[32] Benilan, P., Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M. and Vazquez, J. L.: Existence and uniqueness of solutions of nonlinear elliptic equations in divergence form, To appear.
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