Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 951-967, 2012

Impulsive boundary value problems for $p(t)$-Laplacian's via critical point theory

Marek Galewski, Donal O'Regan

Marek Galewski, Institute of Mathematics, Technical University of Łódź, Wolczanska 215, 90-924 Łódź, Poland, e-mail: marek.galewski@p.lodz.pl; Donal O'Regan, School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland, e-mail: donal.oregan@nuigalway.ie

Abstract: In this paper we investigate the existence of solutions to impulsive problems with a $p(t)$-Laplacian and Dirichlet boundary value conditions. We introduce two types of solutions, namely a weak and a classical one which coincide because of the fundamental lemma of the calculus of variations. Firstly we investigate the existence of solution to the linear problem, i.e. a problem with a fixed rigth hand side. Then we use a direct variational method and next a mountain pass approach in order to get the existence of at least one weak solution to the nonlinear problem.

Keywords: $p(t)$-Laplacian, impulsive condition, critical point, variational method, Dirichlet problem

Classification (MSC 2010): 34B37, 47J30


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