Can't do this because the <% ... %> is run on the
server while the var  j = "" is run on the client.

In other words, one is java while the other is
javascript.

j is just a string and your only choice is to send it to the server side
like so.

<form name="foo" method="post">
<input type="hidden" name="var_j">
<input type="submit">
<script>document.var_j = j</script>

After submitting the form you can assign  it on the server side.
<% String s = request.getParameter("var_j")%>

This will work,

<%
String s="JSP";
%>
<script>
var j="Java Script";
<%=s%>=j
alert(<%=s%>)
</script>