Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 117-158.

We investigate the following quasilinear and singular problem,

to2.7cm-Δ p u=λ u δ +u q inΩ;u| Ω =0,u>0inΩ,to2.7cm(P)
where Ω is an open bounded domain with smooth boundary, 1<p<, p-1<qp * -1, λ>0, and 0<δ<1. As usual, p * =Np N-p if 1<p<N, p * (p,) is arbitrarily large if p=N, and p * = if p>N. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1,p (Ω). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) in C 1,β (Ω ¯) with some β(0,1). Furthermore, we show that δ<1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) in C 1 (Ω ¯).

Classification : 35J65, 35J20, 35J70
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     title = {Sobolev versus {H\"older} local minimizers and existence of multiple solutions for a singular quasilinear equation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {117--158},
     publisher = {Scuola Normale Superiore, Pisa},
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Giacomoni, Jacques; Schindler, Ian; Takáč, Peter. Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 117-158. http://archive.numdam.org/item/ASNSP_2007_5_6_1_117_0/

[1] Adimurthi and J. Giacomoni 8 (2006), 621-656. | MR | Zbl

[2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convexe nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519-543. | MR | Zbl

[3] A. Ambrosetti, J. P. García Azorero and I. Peral Alonso, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. | MR | Zbl

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

[5] A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C.R. Acad. Sci. Paris, Sér. I-Math. 305 (1987), 725-728. | MR | Zbl

[6] A. Anane, “Etude des valeurs propres et de la résonance pour l’opérateur p-Laplacien", Thèse de doctorat, Université Libre de Bruxelles, 1988, Brussels.

[7] C. Aranda and T. Godoy, Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator, Electron. J. Differential Equations 132 (2004), 1-15. | MR | Zbl

[8] F. V. Atkinson and L. A. Peletier, Emden-Fowler equations involving critical exponents, Nonlinear Anal. 10 (1986), 755-776. | MR | Zbl

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

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

[11] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equation involving the critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437-477. | MR | Zbl

[12] H. Brezis and L. Nirenberg, Minima locaux relatifs à C 1 et H 1 , C.R. Acad. Sci. Paris, Sér. I-Math. 317 (1993), 465-472. | Zbl

[13] M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations 14 (1989), 1315-1327. | MR | Zbl

[14] M. Cuesta and P. Takáč, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (2000), 721-746. | MR | Zbl

[15] M. G. Crandall, P. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193-222. | MR | Zbl

[16] K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, Berlin-Heidelberg-New York, 1985. | MR | Zbl

[17] J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris, Sér. I-Math. 305 (1987), 521-524. | MR | Zbl

[18] J. I. Díaz, J. M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations 12 (1987), 1333-1344. | MR | Zbl

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

[20] J. P. García Azorero and I. Peral Alonso, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 (1994), 941-957. | MR | Zbl

[21] J. P. García Azorero, I. Peral Alonso and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), 385-404. | MR | Zbl

[22] J. Giacomoni and K. Sreenadh, Multiplicity results for a singular and quasilinear equation, submitted for publication. | Zbl

[23] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Annals of Mathematics Studies, Princeton University Press, Princeton, N.J., 1983. | MR | Zbl

[24] M. Giaquinta and E. Giusti, Global C 1+α -regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math. 351 (1984), 55-65. | MR | Zbl

[25] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 321-330. | Numdam | MR | Zbl

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

[27] M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879-902. | MR | Zbl

[28] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, New-York, 1983. | MR | Zbl

[29] Y. Haitao, Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), 487-512. | MR | Zbl

[30] J. Hernández, F. Mancebo and J. M. Vega, On the linearization of some singular, nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 777-813. | Numdam | MR | Zbl

[31] N. Hirano, C. Saccon and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations 9 (2004), 197-220. | MR

[32] A. C. Lazer and P. J. Mckenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc. 111 (1991), 721-730. | MR | Zbl

[33] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. | MR | Zbl

[34] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. | MR | Zbl

[35] Z. Nehari, On a class of nonlinear second order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101-123. | MR | Zbl

[36] R. R. Phelps, “Convex Functions, Monotone Operators, and Differentiability”, Lecture notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin, 1993. | MR | Zbl

[37] S. Prashanth and K. Sreenadh, Multiplicity Results in a ball for p-Laplace equation in a ball with positive nonlinearity, Adv. Differential Equations 7 (2002), 877-896. | MR | Zbl

[38] I. Schindler, Quasilinear elliptic boundary-value problems on unbounded cylinders and a related mountain-pass lemma, Arch. Rational Mech. Anal. 120 (1992), 363-374. | MR | Zbl

[39] J. B. Serrin, Local behavior of solutions of quasilinear elliptic equations, Acta Math. 111 (1964), 247-302. | MR | Zbl

[40] P. Takáč, On the Fredholm alternative for the p-Laplacian at the first eigenvalue, Indiana Univ. Math. J. 51 (2002), 187-237. | MR | Zbl

[41] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domans with conical boundary points, Comm. Partial Differential Equations 8 (1983), 773-817. | MR | Zbl

[42] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126-150. | MR | Zbl

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

[44] S. Yijing, W. Shaoping and L. Yiming, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2001), 511-531. | MR | Zbl