An application of variational methods to Dirichlet boundary value problem with impulses

Abstract

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this paper, we deal with the existence and multiplicity of solutions for the nonlinear Dirichlet value problem with impulses. Using the variational methods and critical points theory, we give some new criteria to guarantee that the impulsive problem has at least one nontrivial solution, assuming that the nonlinearity is superquadratic at infinity, subquadratic at the origin, and the impulsive functions have sublinear growth. Moreover, if the nonlinearity and the impulsive functions are odd, then the impulsive problem has infinitely many distinct solutions. Recent results in the literature are generalized and significantly improved.