Applications of variational methods to boundary-value problem for impulsive differential equations. (English) Zbl 1163.34015

Tian, Yu; Ge, Weigao

Proc. Edinb. Math. Soc., II. Ser. 51, No. 2, 509-527 (2008).

The authors study the Sturm-Liouville boundary value problem for a second order impulsive ordinary differential equation of \(p\)-Laplacian type. The aim of this paper is to provide the framework in which the impulsive variational problems can be considered. Since the variational approach to impulsive problems has not yet been developed many difficulties such as the construction of a suitable action functional, the proper function spaces in which the functional has suitable properties, must be overcome.
The authors successfully construct the action functional and investigate the correspondence between its critical points and the classical solutions, in a sense defined in the paper. They also investigate the properties of the space in which they look for the solution. Such results provide the background for further investigation of variational impulsive problems and these are of independent interest.
The main results of this research concerns the existence of at least two positive classical solutions obtained by the Mountain Pass Approach reflecting the appearance of impulses in the considered problem. Examples are given throughout the paper.

Reviewer: Marek Galewski (Łódź)

MSC:

34B37	Boundary value problems with impulses for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
58E30	Variational principles in infinite-dimensional spaces
34B24	Sturm-Liouville theory

Keywords:

Sturm-Liouville boundary value problem; impulsive effects; variational method; mountain-pass theorem