Existence, non-existence and regularity of radial ground states for p-laplacian equations with singular weights
Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 3, pp. 505-537.
@article{AIHPC_2008__25_3_505_0,
     author = {Pucci, Patrizia and Servadei, Raffaella},
     title = {Existence, non-existence and regularity of radial ground states for $p$-laplacian equations with singular weights},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {505--537},
     publisher = {Elsevier},
     volume = {25},
     number = {3},
     year = {2008},
     doi = {10.1016/j.anihpc.2007.02.004},
     zbl = {1147.35045},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2007.02.004/}
}
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Pucci, Patrizia; Servadei, Raffaella. Existence, non-existence and regularity of radial ground states for $p$-laplacian equations with singular weights. Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 3, pp. 505-537. doi : 10.1016/j.anihpc.2007.02.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2007.02.004/

[1] Abdellaoui B., Felli V., Peral I., Existence and non-existence results for quasilinear elliptic equations involving the p-Laplacian, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 9 (2) (2006) 445-484. | MR | Zbl

[2] Adimurthi , Chaudhuri N., Ramaswamy M., An improved Hardy-Sobolev inequality and its application, Proc. Amer. Math. Soc. 130 (2) (2001) 489-505. | MR | Zbl

[3] Ambrosetti A., Rabinowitz P., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381. | MR | Zbl

[4] Berestycki H., Lions J.L., Nonlinear scalar field equations, I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983) 313-345. | MR | Zbl

[5] Berestycki H., Lions J.L., Nonlinear scalar field equations, II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983) 347-375. | MR | Zbl

[6] Boccardo L., Murat F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., Theory Methods Appl. 19 (6) (1992) 581-597. | MR | Zbl

[7] Brézis H., Analyse Fonctionelle. Théorie et applications, Masson, Paris, 1983. | MR | Zbl

[8] Brézis H., Lieb E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (3) (1983) 486-490. | MR | Zbl

[9] Caffarelli L., Kohn R., Nirenberg L., First order inequalities with weights, Compos. Math. 53 (1984) 259-275. | Numdam | MR | Zbl

[10] Caldiroli P., Malchiodi A., Singular elliptic problems with critical growth, Commun. Partial Differential Equations 27 (2002) 847-876. | MR | Zbl

[11] Calzolari E., Filippucci R., Pucci P., Existence of radial solutions for the p-Laplacian elliptic equations with weights, Discrete Contin. Dyn. Syst. 15 (2) (2006) 447-479. | MR | Zbl

[12] Chaudhuri N., Ramaswamy M., Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proc. R. Soc. Edinburgh Sect. A Math. 131 (6) (2001) 1275-1295. | MR | Zbl

[13] Citti G., Positive solutions for a quasilinear elliptic equation in R n , Rend. Circ. Mat. Palermo, II. Ser. 35 (1986) 364-375. | MR | Zbl

[14] Clément Ph., Manásevich R., Mitidieri E., Some existence and non-existence results for a homogeneous quasilinear problem, Asymptotic Anal. 17 (1998) 13-29. | MR | Zbl

[15] Coleman S., Glaser V., Martin A., Action minima among solutions to a class of euclidean scalar field equations, Comm. Math. Phys. 58 (1978) 211-221. | MR

[16] Dibenedetto E., C 1+α -local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., Theory Methods Appl. 7 (8) (1983) 827-850. | MR | Zbl

[17] Ekeland I., Ghoussoub N., Selected new aspects of the calculus of variations in the large, Bull. Am. Math. Soc. (N.S) 39 (2) (2002) 207-265. | MR | Zbl

[18] Ferrero A., Gazzola F., Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001) 494-522. | MR | Zbl

[19] Ferrero A., Gazzola F., On subcriticality assumptions for the existence of ground states of quasilinear elliptic equations, Adv. Differential Equations 8 (2003) 1081-1106. | MR

[20] García Azorero J.P., Peral I., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441-476. | MR | Zbl

[21] García Huidobro J.P., Manásevich R., Yarur C., On the structure of positive radial solutions to an equation containing a p-Laplacian with weight, J. Differential Equations 223 (2006) 51-95. | MR | Zbl

[22] Gazzola F., Serrin J., Tang M., Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. Differential Equations 5 (1-3) (2000) 1-30. | MR | Zbl

[23] N. Ghoussoub, F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap. (2006), 21867, 1-85. | MR | Zbl

[24] Ghoussoub N., Yuan C., Multiple solutions for quasilinear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (12) (2000) 5703-5743. | MR | Zbl

[25] Goncalves J.V., Santos C.A.P., Positive solutions for a class of quasilinear singular equations, Electron. J. Differential Equations 2004 (56) (2004) 1-15. | MR | Zbl

[26] Hewitt E., Stromberg K., Real and Abstract Analysis, Springer-Verlag, Berlin, 1965. | MR | Zbl

[27] E. Jannelli, S. Solimini, Critical behaviour of some elliptic equations with singular potentials, Rapporto n. 41, Università degli Studi di Bari, 1996.

[28] Kavian O., Introdution à la théorie des points critiques, Springer-Verlag, Paris, 1983.

[29] Kufner A., Persson L.E., Weighted Inequalities of Hardy-Type, Word Scientific, 2003. | MR | Zbl

[30] Lay D.C., Taylor A.E., Introduction to Functional Analysis, John Wiley and Sons, New York, 1980. | MR | Zbl

[31] Li J., Equation with critical Sobolev-Hardy exponents, Int. J. Math. Math. Sci. 20 (2005) 3213-3223. | MR | Zbl

[32] Lieb E.H., Loss M., Analysis, Graduate Studies in Math., vol. 14, second ed., Amer. Math. Soc., 1997. | MR | Zbl

[33] Lions P.L., The concentration-compactness principle in the calculus of variations. The limit case, Part 1, Rev. Mat. Iberoamericana 1 (1985) 145-201. | MR | Zbl

[34] Mitidieri E., A simple approach to Hardy inequalities, Mat. Zametki 67 (2000) 563-572, (in Russian); translation in, Math. Notes 67 (2000) 479-486. | MR | Zbl

[35] Ni W.-M., Serrin J., Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) 8 (Centenary Supplement) (1985) 171-185. | Zbl

[36] Pucci P., García-Huidobro M., Manásevich R., Serrin J., Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl. (4) 185 (2006) 205-243. | MR | Zbl

[37] Pucci P., Serrin J., Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (1998) 501-528. | MR | Zbl

[38] P. Pucci, J. Serrin, The Strong Maximum Principle, Progress in Nonlinear Differential Equations, Birkhäuser Publ., Switzerland, monograph book, pp. 206, in press. | MR | Zbl

[39] Pucci P., Serrin J., Zou H., A strong maximum principle and a compact support principle for singular elliptic inequalities, J. Math. Pures Appl. (9) 78 (1999) 769-789. | MR | Zbl

[40] Pucci P., Servadei R., On weak solutions for p-Laplacian equations with weights, Rend. Lincei Mat. Appl. 18 (2007) 257-267. | MR

[41] P. Pucci, R. Servadei, Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations, submitted for publication. | Zbl

[42] Ruiz D., Willem M., Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations 190 (2003) 524-538. | MR | Zbl

[43] Strauss W.A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) 149-162. | MR | Zbl

[44] Struwe M., Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, third ed., Springer-Verlag, Berlin, 2000. | MR | Zbl

[45] Swanson C.A., Yu L.S., Critical p-Laplacian problems in R n , Ann. Mat. Pura Appl. (4) 169 (1995) 233-250. | MR | Zbl

[46] Talenti G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353-372. | MR | Zbl

[47] Terracini S., On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (2) (1996) 241-264. | MR | Zbl

[48] Vázquez J.L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization 12 (1984) 191-202. | MR | Zbl

[49] Willem M., Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996. | MR | Zbl

[50] Xuan B., The solvability of quasilinear Brézis-Nirenberg-type problems wit singular weights, Nonlinear Anal., Theory Methods Appl. 62 (2005) 703-725. | MR | Zbl

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