Existence and nonexistence of positive solutions for quasilinear elliptic problem. (English) Zbl 1250.35086

Summary: Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving the \(p\)-Laplacian.


35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
Full Text: DOI


[1] V. R\uadulescu and D. Repov\vs, “Combined effects in nonlinear problems arising in the study of anisotropic continuous media,” Nonlinear Analysis, vol. 75, no. 3, pp. 1524-1530, 2012. · Zbl 1237.35043 · doi:10.1016/j.na.2011.01.037
[2] C. A. Santos, “Non-existence and existence of entire solutions for a quasi-linear problem with singular and super-linear terms,” Nonlinear Analysis, vol. 72, no. 9-10, pp. 3813-3819, 2010. · Zbl 1189.35104 · doi:10.1016/j.na.2010.01.017
[3] M. Cuesta and P. Taká\vc, “A strong comparison principle for positive solutions of degenerate elliptic equations,” Differential and Integral Equations, vol. 13, no. 4-6, pp. 721-746, 2000. · Zbl 0973.35077
[4] P. Lindqvist, “On the equation div(|\nabla u|p-2\nabla u)+\lambda |u|p-2u=0,” Proceedings of the American Mathematical Society, vol. 109, no. 1, pp. 157-164, 1990. · Zbl 0714.35029 · doi:10.2307/2048375
[5] A. Anane, “Simplicité et isolation de la première valeur propre du p-laplacien avec poids,” Comptes Rendus des Séances de l’Académie des Sciences I, vol. 305, no. 16, pp. 725-728, 1987. · Zbl 0633.35061
[6] R. Filippucci, P. Pucci, and V. R\uadulescu, “Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions,” Communications in Partial Differential Equations, vol. 33, no. 4-6, pp. 706-717, 2008. · Zbl 1147.35038 · doi:10.1080/03605300701518208
[7] P. Pucci and R. Servadei, “Regularity of weak solutions of homogeneous or inhomogeneous quasilinear elliptic equations,” Indiana University Mathematics Journal, vol. 57, no. 7, pp. 3329-3363, 2008. · Zbl 1171.35057 · doi:10.1512/iumj.2008.57.3525
[8] E. DiBenedetto, “C1+\alpha local regularity of weak solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 7, no. 8, pp. 827-850, 1983. · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5
[9] P. Pucci and J. Serrin, “Maximum principles for elliptic partial differential equations,” in Handbook of Differential Equations: Stationary Partial Differential Equations, M. Chipot, Ed., vol. 4, pp. 355-483, Elsevier, 2007. · Zbl 1193.35024 · doi:10.1016/S1874-5733(07)80009-X
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