Finally i got a solution for this problem.I were removed global variable and changed function ShowMe
name to ShowMe<%=this.ClientID%>
and also update ShowMe<%=this.ClientID%>
in delegate.And it works.
Updated Code
function DropDownExtender1_pageLoad() {
var DDE;
DDE = $find('<%= DDE.ClientID%>');
DDE._dropWrapperHoverBehavior_onhover();
$get('<%= DropPanel.ClientID%>').style.width = $get('<%= txtDisplay.ClientID%>').clientWidth;
if (DDE._dropDownControl) {
$common.removeHandlers(DDE._dropDownControl, DDE._dropDownControl$delegates);
}
DDE._dropDownControl$delegates = {
click: Function.createDelegate(DDE, ShowMe<%=this.ClientID %>),
contextmenu: Function.createDelegate(DDE, DDE._dropDownControl_oncontextmenu)
}
$addHandlers(DDE._dropDownControl, DDE._dropDownControl$delegates);
//dropdown arrow key allways visible.
$find('<%= DDE.ClientID%>').unhover = doNothing;
$find('<%= DDE.ClientID%>')._dropWrapperHoverBehavior_onhover();
//Maintain scroll position
var h = document.getElementById('<%= hfScrollPosition.ClientID%>');
document.getElementById('<%= divGrid.ClientID%>').scrollTop = h.value;
}
//do nothing on arrow key hover
function doNothing() { }
//Ragister onload event
Sys.Application.add_load(DropDownExtender1_pageLoad);
//Ragister on page load
function ShowMe<%=this.ClientID %>() {
var DDE = $find('<%= DDE.ClientID%>');
DDE._wasClicked = true;
}