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 {\protect\emph{p}-Laplacian} involving critical and supercritical nonlinearities with zeros}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {763--771}, publisher = {Elsevier}, volume = {27}, number = {2}, year = {2010}, doi = {10.1016/j.anihpc.2009.11.003}, mrnumber = {2595200}, zbl = {1187.35096}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.003/} }
TY - JOUR AU - Iturriaga, Leonelo AU - Lorca, Sebastián AU - Massa, Eugenio TI - Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 763 EP - 771 VL - 27 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.003/ DO - 10.1016/j.anihpc.2009.11.003 LA - en ID - AIHPC_2010__27_2_763_0 ER -
%0 Journal Article %A Iturriaga, Leonelo %A Lorca, Sebastián %A Massa, Eugenio %T Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 763-771 %V 27 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.003/ %R 10.1016/j.anihpc.2009.11.003 %G en %F 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://archive.numdam.org/articles/10.1016/j.anihpc.2009.11.003/
[1] A. Anane, Etude des valeurs propres et de la résonnance pour l'opérateur p-Laplacien, PhD thesis, Universit Libre de Bruxelles, 1987
[2] On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 no. 1 (1991), 1-37 | MR | Zbl
, ,[3] On a nonlinear elliptic eigenvalue problem, J. Math. Anal. Appl. 307 no. 2 (2005), 691-698 | MR | Zbl
, ,[4] On pairs of positive solutions for a class of semilinear elliptic problems, Indiana Univ. Math. J. 34 no. 3 (1985), 591-606 | Zbl
, ,[5] Local “superlinearity” and “sublinearity” for the p-Laplacian, J. Funct. Anal. 257 no. 3 (2009), 721-752 | MR | Zbl
, , ,[6] Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 no. 2 (2004), 483-515 | MR | Zbl
, ,[7] Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations 13 no. 10–12 (2000), 1201-1232 | MR | Zbl
, ,[8] Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 no. 8 (1989), 879-902 | MR | Zbl
, ,[9] L. Iturriaga, S. Lorca, M. Montenegro, Existence of solutions to quasilinear elliptic equations with singular weights, Adv. Nonlinear Stud., in press | MR
[10] 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 | Zbl
, , ,[11] Positive solutions for the p-Laplacian with a nonlinear term with zeros, J. Differential Equations 248 no. 2 (2010), 309-327 | MR | Zbl
, , , ,[12] Flat core properties associated to the p-Laplace operator, Proc. Amer. Math. Soc. 118 no. 4 (1993), 1079-1085 | MR | Zbl
, ,[13] Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203-1219 | MR | Zbl
,[14] On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 no. 4 (1982), 441-467 | MR | Zbl
,[15] Nonexistence of positive solution for quasilinear elliptic problems in the half-space, J. Inequal. Appl. (2007) | EuDML | MR | Zbl
,[16] Partial differential equations involving subcritical, critical and supercritical nonlinearities, Nonlinear Anal. 56 no. 1 (2004), 119-131 | MR | Zbl
, ,[17] On the eigenfunctions of the equation , Dokl. Akad. Nauk SSSR 165 (1965), 36-39 | MR
,[18] A general variational identity, Indiana Univ. Math. J. 35 no. 3 (1986), 681-703 | MR | Zbl
, ,[19] Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 no. 1 (2002), 79-142 | MR | Zbl
, ,[20] 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 | Zbl
,[21] On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 no. 7 (1983), 773-817 | MR | Zbl
,[22] Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 no. 1 (1984), 126-150 | MR | Zbl
,[23] A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 no. 3 (1984), 191-202 | MR | Zbl
,Cited by Sources: