Thanks! Yes, I can see now that searching works. The reason I missed it was because the Page title I was searching for didn't look as expected. Instead of the title "Försök", your select box showed the title as "<mark class='searchwp-highlight'>Försök</mark>". Yes I use the plugin SearchWP. But if I go to Pages, the title shows up as "Försök" and in DB the field "post_title" in table "wp_posts" is also "Försök".
So, why is the <mark>-code showing up in your Select box?