Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

Abstract EN

This paper is concerned with the following nonlinear second-order nonautonomous problem: u ̈ ( t ) + q ( t ) u ̇ ( t ) - ∇ K ( t , u ( t ) ) + ∇ W ( t , u ( t ) ) = 0 , where t ∈ R , u ∈ R N , and K , W ∈ C 1 ( R × R N , R ) are not periodic in t and q : R → R is a continuous function and Q ( t ) = ∫ 0 t q ( s ) d s with lim | t | → + ∞ Q ( t ) = + ∞ .

The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.