Multiple Results of Damped Systems with General Nonlinearities

Abstract

Abstract We study a class of non-periodic damped systems, where the nonlinearity H is subquadratic at 0 and general at infinity (i.e., it can be subquadratic, asymptotically quadratic or superquadratic at infinity, three examples are constructed to illustrate the three different cases). Besides, the global evenness condition ( H ⁢ ( t , - u ) = H ⁢ ( t , u ) ${H(t,-u)=H(t,u)}$ ) and the global non-negative condition ( H ⁢ ( t , u ) ≥ 0 ${H(t,u)\geq 0}$ ) used before are, respectively, replaced by the local evenness condition ( H ⁢ ( t , - u ) = H ⁢ ( t , u ) ${H(t,-u)=H(t,u)}$ , | u | ≤ r ${|u|\leq r}$ ) and the local non-negative condition ( H ⁢ ( t , u ) ≥ 0 ${H(t,u)\geq 0}$ , | u | ≤ r ${|u|\leq r}$ ). Under the above weaker and more general conditions, we obtain infinitely many nontrivial small negative energy homoclinic orbits for this class of damped systems by an extension of Clark’s theorem.