SIGN-CHANGING SOLUTIONS FOR A CLASS OF DAMPED VIBRATION PROBLEMS WITH IMPULSIVE EFFECTS

Abstract

In this paper, some sufficient conditions are obtained for theexistence and multiplicity of sign-changing solutions for the dampedvibration problem with impulsive effects \begin{eqnarray*} \left\{ \begin{array}{ll} -u''(t)+g(t)u'(t)=f(t,u(t)), & \hbox{a.e. $t\in [0,T]$;} \\ u(0)=u(T)=0, & \hbox{} \\ \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})), & \hbox{$j=1,2,\ldots,p,$} \end{array} \right. \end{eqnarray*} where $t_{0}=0\lt t_{1}\lt t_{2}\lt \ldots\lt t_{p}\lt t_{p+1}=T,g\in L^{1}(0,T;\mathbb{R}),I_{j}:\mathbb{R}\rightarrow\mathbb{R},j=1,2,\ldots,p$ are continuous, $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function with subcritical growth condition:

$(A) |f(t, u)| ≤ C(1 + |u|^{s-1}), \forall t \in[0,T], u\in \mathbb{R}, s\in [2,+\infty)$.

The sign-changing solutions are sought by means of some sign-changing critical point theorems and two examples are presented to illustrate the feasibility and effectiveness of our results.