Abstract
In this paper, the authors study the existence and non-existence of positive solutions for singular p-Laplacian equation −∆ p u = f(x)u −α + λg(x)u β in R N; where N ≥ 3, 1 < p < N, λ > 0, 0 < α < 1, max(p, 2) < β + 1 < p* = \( \frac{{{N_p}}}{{N - p}} \). We prove that there exists a critical value ¤ such that the problem has at least two solutions if 0 < λ < Λ; at least one solution if λ = Λ; and no solutions if λ > Λ.
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M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 1977, 2: 193–232.
A. V. Lair and A. W. Shaker, Classical and weak solutions of a singular semilinear elliptic problem, J. Math. Anal. Appl., 1997, 211: 193–222.
A. C. Lazer and P. J. Mckenna, On a singular nonlinear elliptic boundary value problem, in Proc. Amer. Math. Aoc., 1191, 111: 721–730.
M. A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, in Proc. Roy. Soc. Edinburgh Sect. A, 1992, 122: 341–352.
J. I. Diaz, J. M. Morel, and I. Oswald, An elliptic equation with singular nonlinearity, Comm. Partial Differential Equations, 1987, 12: 1333–1344.
M. M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet poblem, Comm. Partial Differential Equations, 1989, 14: 1315–1327.
H. T. Yang, Muitiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem, J. Differential EquAtions, 2003, 189: 487–512.
J. Hernandez, F. J. Mancebo, and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, in Proc. Roy. Soc. Edinburgh, 2007, 137: 41–62.
H. Hiraro and C. Saccon, and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with a concave and convex nonlinearities, Adv. Differential Equations, 2004, 9: 197–220.
Y. Sun and S. Li, Structure of ground state solutions of singular elliptic equations, Nonlinear analysis, 2003, 55: 399–417.
A. Canino and M. Degiovanni, Nonsmooth critical point theory and quasilinear elliptic equations, in Topological Methods in Differential Equations and Inclusions, Montreal, 1994, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 472, 1–50.
M. Choulli, R. Deville, and A. Rhandi, A general mountain pass principle for nondifferential functionals, Rev. Mat. Apl., 1992, 13: 45–58.
L. N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl., 1995, 196: 1050–1072.
M. Degiovanni and M. Marzocchi, A critical point theory for n onsmooth functional, Ann. Math. Pura. Appl., 1994, 167: 73–100.
D. G. Costa and J. V. A. Goncalves, Critical point theory for nondifferential functionals and applications, J. Math. Anal. Appl., 1990, 153: 470–485.
Y. Guo and J. Liu, Critical point theory for non smooth functionals, Nonlinear Analysis, 2007, 66: 2731–2741.
P. Lindqvist, On the equation div(|∇u|p−2∇u) + λ|u|p−2 u = 0; in Proc. Amer. Math. Soc., 1990, 109: 157–163.
N. S. Tridinger, On Harnack type inequalities and their applications to quasilinear equations, Comm. Pure. Appl.Math., 1967, 20: 721–747.
N. Hirano, C. Saccon, and N. Shioji, Brezis Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Diff. Equn., 2008, 245: 1997–2037.
A. Edelson, Entire solutions of singular elliptic equations, J. Math. Ana. Appl., 1989, 139: 523–532.
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This research is supported by Natural Science Foundation of China under Grant No. 10871110.
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Liu, X., Guo, Y. & Liu, J. Solutions for singular p-Laplacian equation in R n . J Syst Sci Complex 22, 597–613 (2009). https://doi.org/10.1007/s11424-009-9190-6
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DOI: https://doi.org/10.1007/s11424-009-9190-6