Previous |  Up |  Next


critical point theory; lower and upper solutions; impulsive; $p$-Laplacian
Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a $p$-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the $p$-Laplacian impulsive problem.
[1] Anello, G., Cordaro, G.: Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the $p$-Laplacian. Proc. R. Soc. Edinb., Sect. A 132 (2002), 511-519. DOI 10.1017/S030821050000175X | MR 1912413
[2] Cîrstea, F., Motreanu, D., Rădulescu, V.: Weak solutions of quasilinear problems with nonlinear boundary condition. Nonlinear Anal., Theory Methods Appl. 43 (2001), 623-636. DOI 10.1016/S0362-546X(99)00224-2 | MR 1804861
[3] Costa, D. G., Magalhães, C. A.: Existence results for perturbations of the $p$-Laplacian. Nonlinear Anal., Theory Methods Appl. 24 (1995), 409-418. DOI 10.1016/0362-546X(94)E0046-J | MR 1312776
[4] Coster, C. De, Habets, P.: Two-point Boundary Value Problems. Lower and Upper Solutions. Elsevier Amsterdam (2006). MR 2225284
[5] Amrouss, A. R. El, Moussaoui, M.: Minimax principle for critical point theory in applications to quasilinear boundary value problems. Electron. J. Differ. Equ. 18 (2000), 1-9. MR 1744087
[6] Guo, Y., Liu, J.: Solutions of $p$-sublinear $p$-Laplacian equation via Morse theory. J. Lond. Math. Soc. 72 (2005), 632-644. DOI 10.1112/S0024610705006952 | MR 2190329 | Zbl 1161.35405
[7] Nieto, J. J., O'Regan, D.: Variational approach to impulsive differential equation. Nonlinear Anal., Real World Appl. 10 (2009), 680-690. MR 2474254
[8] Omari, P., Zanolin, F.: An elliptic problem with arbitrarily small positive solutions. Electron. J. Differ. Equ., Conf. 05 (2000), 301-308. MR 1799060 | Zbl 0959.35059
Partner of
EuDML logo