Absorption effects for some elliptic equations with singularities

A. Porretta

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 2, page 369-395
  • ISSN: 0392-4041

Abstract

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We give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data. As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6).

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Porretta, A.. "Absorption effects for some elliptic equations with singularities." Bollettino dell'Unione Matematica Italiana 8-B.2 (2005): 369-395. <http://eudml.org/doc/195539>.

@article{Porretta2005,
abstract = {We give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data. As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6).},
author = {Porretta, A.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {369-395},
publisher = {Unione Matematica Italiana},
title = {Absorption effects for some elliptic equations with singularities},
url = {http://eudml.org/doc/195539},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Porretta, A.
TI - Absorption effects for some elliptic equations with singularities
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/6//
PB - Unione Matematica Italiana
VL - 8-B
IS - 2
SP - 369
EP - 395
AB - We give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data. As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6).
LA - eng
UR - http://eudml.org/doc/195539
ER -

References

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  1. BANDLE, C. - GIARRUSSO, E., Boundary blow-up for semilinear elliptic equations with nonlinear gradient terms, Adv. Diff. Equat., 1 (1996), 133-150. Zbl0840.35034MR1357958
  2. BANDLE, C. - MARCUS, M., Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Anal. Math., 58 (1992), 9-24. Zbl0802.35038MR1226934
  3. BENILAN, P. - BOCCARDO, L. - GALLOUËT, T. - GARIEPY, R. - PIERRE, M. - VÁZQUEZ, J. L., An L 1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 240-273. Zbl0866.35037MR1354907
  4. BENILAN, P. - BREZIS, H., Nonlinear problems related to the Thomas-Fermi equation, J. Evolution Equations, 3 (2003), 673-770. Zbl1150.35406MR2058057
  5. BENSOUSSAN, A. - BOCCARDO, L. - MURAT, F., On a nonlinear partial differential equation having natural growth terms and unbounded solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 347-364. Zbl0696.35042MR963104
  6. BOCCARDO, L., T-minima: an approach to minimization problems in L 1 , Contributions in honor of the memory of Ennio De Giorgi. Ricerche Mat., 49 (2000), 135-154. Zbl1009.49002MR1826220
  7. BOCCARDO, L. - GALLOUËT, T., Strongly nonlinear elliptic equations having natural growth terms and L 1 data, Nonlinear Anal. T.M.A., 19 (1992), 573-579. Zbl0795.35031MR1183664
  8. BOCCARDO, L. - GALLOUËT, T. - ORSINA, L., Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551. Zbl0857.35126MR1409661
  9. BOCCARDO, L. - GALLOUËT, T. - ORSINA, L., Existence and nonexistence of solutions for some nonlinear elliptic equations, Journal d'Analyse Math., 73 (1997), 203-223. Zbl0898.35035MR1616410
  10. BREZIS, H., Nonlinear elliptic equations involving measures, in Variational Inequalities, Cottle, Giannessi, Lions ed., Wiley, 1980, 53-73. Zbl0643.35108
  11. BREZIS, H., Some Variational Problems of the Thomas-Fermi type, in Contributions to nonlinear partial differential equations (Madrid, 1981), 82-89, Res. Notes in Math.89, Pitman, Boston Mass.-London, 1983. Zbl0533.35038MR578739
  12. BREZIS, H., Semilinear equations in R N without condition at infinity, Appl. Math. Optim., 12, no. 3 (1984), 271-282. Zbl0562.35035MR768633
  13. BREZIS, H. - MARCUS, M. - PONCE, A., Nonlinear elliptic equations with measures revisited, to appear in Annals of Math. Studies, Princeton Univ. Press. Zbl1151.35034MR2333208
  14. BREZIS, H. - NIRENBERG, L., Removable singularities for some nonlinear elliptic equations, Top. Methods Nonlin. Anal., 9 (1997), 201-219. Zbl0905.35027MR1491843
  15. DAL MASO, G. - MURAT, F. - ORSINA, L. - PRIGNET, A., Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28, 4 (1999), 741-808. Zbl0958.35045MR1760541
  16. FUKUSHIMA, M. - SATO, K. - TANIGUCHI, S., On the closable part of pre-Dirichlet forms and the fine support of the underlying measures, Osaka J. Math., 28 (1991), 517-535. Zbl0756.60071MR1144471
  17. GMIRA, A. - VERON, L., Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J., 64 (1991), 271-324. Zbl0766.35015MR1136377
  18. IWANIEC, T. - SBORDONE, C., Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. Zbl0802.35016MR1288682
  19. KELLER, J. B., On solutions of Δ u = f u , Comm. Pure Appl. Math., 10 (1957), 503-510. Zbl0090.31801MR91407
  20. LASRY, J.-M. - LIONS, P.-L., Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., 283, n. 4 (1989), 583-630. Zbl0688.49026MR990591
  21. LERAY, J. - LIONS, J.-L., Quelques resultats de Višik sur les problemes elliptiques non lineaires par les methodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. Zbl0132.10502MR194733
  22. LOEWNER, C. - NIRENBERG, L., Partial differential equations invariant under conformal or projective transformations, Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245-272, Academic Press, New York, 1974. Zbl0298.35018MR358078
  23. MARCUS, M. - VERON, L., Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 14 (1997), 237-274. Zbl0877.35042MR1441394
  24. MARCUS, M. - VERON, L., Existence and uniqueness results for large solutions of general nonlinear elliptic equations, Journal of Evolution Equations, 3 (2003), 637-652. Zbl1160.35408MR2058055
  25. MURAT, F. - PORRETTA, A., Stability properties, existence and nonexistence of renormalized solutions for elliptic equations with measure data, Comm. P.D.E., 27, n. 11 & 12 (2002), 2267-2310. Zbl1129.35397MR1944030
  26. ORSINA, L., Weak minima for some functionals and elliptic equations involving measures, C. R. Acad. Sci. Paris322, Serie I, (1996), 1151-1156. Zbl0849.35024MR1396657
  27. ORSINA, L. - PORRETTA, A., Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data, Comm. in Contemporary Math., 3 (2001), 259-285. Zbl1162.35371MR1831931
  28. OSSERMAN, R., On the inequality Δ u ; f ( u ) , Pacific J. Math., 7 (1957), 1641-1647. Zbl0083.09402MR98239
  29. PORRETTA, A., Some remarks on the regularity of solutions for a class of elliptic equations with measure data, Houston Journ. of Math., 26 (2000), 183-213. Zbl0974.35032MR1814734
  30. PORRETTA, A., Remarks on the existence or loss of minima of infinite energy, preprint. Zbl1194.49008MR2337027
  31. PORRETTA, A., Local estimates and large solutions for some elliptic equations with absorption, Advances in Diff. Eq.9, n. 3-4 (2004), 329-351. Zbl1150.35401MR2100631
  32. PORRETTA, A., PHD Thesis, Università di Roma «La Sapienza», 1999. 
  33. PORRETTA, A., Existence for elliptic equations in L 1 having lower order terms with natural growth, Portugaliae Math., 57 (2000), 179-190. Zbl0963.35068MR1759814
  34. PORRETTA, A., Some uniqueness results for elliptic equations without condition at infinity, Commun. Contemporary Mathematics, 5, n. 5 (2003), 1-13. Zbl1156.35369MR2017714
  35. STAMPACCHIA, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, (Grenoble), 15, n. 1 (1965), 189-258. Zbl0151.15401MR192177
  36. L. VAZQUEZ, J., On a semilinear equation in R 2 involving bounded measures, Proc. Royal Soc. Edinburgh, 95A (1983), 181-202. Zbl0536.35025MR726870
  37. VERON, L., Semilinear elliptic equations with uniform blow-up on the boundary, J. Analyse Math., 59 (1992), 231-250. Zbl0802.35042MR1226963
  38. VERON, L., Elliptic equations involving measures, to appear in Handbook for Partial Differential Equations, M. Chipot, P. Quittner Eds, Elsevier. Zbl1129.35478MR2103694

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