Marek Galewski, Institute of Mathematics, Technical University of Łódź, Wolczanska 215, 90-924 Łódź, Poland, e-mail: email@example.com; Donal O'Regan, School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland, e-mail: firstname.lastname@example.org
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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