Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, p. 763-771
In this paper we show the existence of multiple solutions to a class of quasilinear elliptic equations when the continuous nonlinearity has a positive zero and it satisfies a p-linear condition only at zero. In particular, our approach allows us to consider superlinear, critical and supercritical nonlinearities.
@article{AIHPC_2010__27_2_763_0,
     author = {Iturriaga, Leonelo and Lorca, Sebasti\'an and Massa, Eugenio},
     title = {Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {2},
     year = {2010},
     pages = {763-771},
     doi = {10.1016/j.anihpc.2009.11.003},
     zbl = {1187.35096},
     mrnumber = {2595200},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_2_763_0}
}
Iturriaga, Leonelo; Lorca, Sebastián; Massa, Eugenio. Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 763-771. doi : 10.1016/j.anihpc.2009.11.003. http://www.numdam.org/item/AIHPC_2010__27_2_763_0/

[1] A. Anane, Etude des valeurs propres et de la résonnance pour l'opérateur p-Laplacien, PhD thesis, Universit Libre de Bruxelles, 1987

[2] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 no. 1 (1991), 1-37 | MR 1159383 | Zbl 0784.35025

[3] S. Chen, S. Li, On a nonlinear elliptic eigenvalue problem, J. Math. Anal. Appl. 307 no. 2 (2005), 691-698 | MR 2142453 | Zbl 1080.35055

[4] D.G. De Figueiredo, P.-L. Lions, On pairs of positive solutions for a class of semilinear elliptic problems, Indiana Univ. Math. J. 34 no. 3 (1985), 591-606 | Zbl 0587.35033

[5] D.G. De Figueiredo, J.-P. Gossez, P. Ubilla, Local “superlinearity” and “sublinearity” for the p-Laplacian, J. Funct. Anal. 257 no. 3 (2009), 721-752 | MR 2530603 | Zbl 1178.35176

[6] L. Damascelli, B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 no. 2 (2004), 483-515 | MR 2096703 | Zbl 1108.35069

[7] J. García-Melián, J. Sabina De Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations 13 no. 10–12 (2000), 1201-1232 | MR 1785705 | Zbl 0976.35021

[8] M. Guedda, L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 no. 8 (1989), 879-902 | MR 1009077 | Zbl 0714.35032

[9] L. Iturriaga, S. Lorca, M. Montenegro, Existence of solutions to quasilinear elliptic equations with singular weights, Adv. Nonlinear Stud., in press | MR 2574376

[10] L. Iturriaga, S. Lorca, J. Sánchez, Existence and multiplicity results for the p-Laplacian with a p-gradient term, NoDEA Nonlinear Differential Equations Appl. 15 no. 6 (2008), 729-743 | MR 2465780 | Zbl 1170.35408

[11] L. Iturriaga, E. Massa, J. Sanchez, P. Ubilla, Positive solutions for the p-Laplacian with a nonlinear term with zeros, J. Differential Equations 248 no. 2 (2010), 309-327 | MR 2558168 | Zbl 1181.35117

[12] S. Kamin, L. Véron, Flat core properties associated to the p-Laplace operator, Proc. Amer. Math. Soc. 118 no. 4 (1993), 1079-1085 | MR 1139470 | Zbl 0801.35029

[13] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203-1219 | MR 969499 | Zbl 0675.35042

[14] P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 no. 4 (1982), 441-467 | MR 678562 | Zbl 0511.35033

[15] S. Lorca, Nonexistence of positive solution for quasilinear elliptic problems in the half-space, J. Inequal. Appl. (2007) | MR 2291650 | Zbl 1200.35142

[16] S. Lorca, P. Ubilla, Partial differential equations involving subcritical, critical and supercritical nonlinearities, Nonlinear Anal. 56 no. 1 (2004), 119-131 | MR 2031438 | Zbl 1232.35061

[17] S.I. Pohožaev, On the eigenfunctions of the equation Δu+λf(u)=0, Dokl. Akad. Nauk SSSR 165 (1965), 36-39 | MR 192184

[18] P. Pucci, J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 no. 3 (1986), 681-703 | MR 855181 | Zbl 0625.35027

[19] J. Serrin, H. Zou, Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 no. 1 (2002), 79-142 | MR 1946918 | Zbl 1059.35040

[20] S. Takeuchi, Partial flat core properties associated to the p-Laplace operator, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference Discrete Contin. Dyn. Syst. no. Suppl. (2007), 965-973 | MR 2409934 | Zbl 1163.35385

[21] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 no. 7 (1983), 773-817 | MR 700735 | Zbl 0515.35024

[22] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 no. 1 (1984), 126-150 | MR 727034 | Zbl 0488.35017

[23] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 no. 3 (1984), 191-202 | MR 768629 | Zbl 0561.35003