Abstract
In this paper, we study the existence and multiplicity of solutions for an impulsive differential equation via some critical point theory and the variational method. We extend and improve some recent results and reduce conditions.
Highlights
1 Introduction As an important research field of study, the impulsive differential equation has been attracting the attention of several mathematicians
The main way to resolve this kind of problems is based on the fixed point theory, the theorem of topological degree, the upper and lower solutions method coupled with the monotone iterative technique, and so on; see for example [ – ]
In [ ], authors have shown the variational structure of an impulsive differential equation and proved the existence of a solution by using the mountain pass lemma
Summary
As an important research field of study, the impulsive differential equation has been attracting the attention of several mathematicians. Many authors have tried to use the variational method and some specific critical point theorems, such as mountain pass lemma, fountain theorem, linking theorem, symmetric mountain pass lemma, and so on, to study the existence (see [ – ]) and multiplicity (see [ – ]) of solutions for some impulsive differential equations. In [ ], authors have shown the variational structure of an impulsive differential equation and proved the existence of a solution by using the mountain pass lemma. P, In this paper, we study the existence and multiplicity of solutions for the following nonlinear impulsive problem:. We will prove that equation ( ) has at least two classical solutions and infinitely many classical solutions under different conditions. We prove the same impulsive problem in [ ] cannot only have two solutions and have infinitely many classical solutions.
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