Does the average family there make enough money to afford sending their child to a program overseas?
Yes, they can take a loan in their house, but...
the authorities would be all over them if the child has left the country. We have a lot of people, who have come to Denmark from the Middle East. In the past some of them shipped their children back, if the children became "too danish" (Married people their parents had not selected for them etc.) Those children returned in a state that made them not suitable to live in Denmark properly. So all traffic with children out of the Denmark is monitored very carefully. If a child suddenly is missing in school, the hell breaks loose. They even called me one morning where I forgot to phone them about my daughters illness. The first day !!!
Schooling out of the country is not an option.
Danish children should learn Danish values. That means free access to alcohol once they have obtained their ID-card, which they can get down at the city hall once they turn 16.
That means that the police only would locate and monitor a runaway. They would not bring it back home unless the child is staying in a house where only adults are present or otherwise endanger itself. It is the parents problem to persuade the child to return home.
Are there boarding schools or something like that there? What is the closest thing to a program in your country?
We have court-ordered programs and programs where the politicians down at the city hall based on recommendation from a social worker can order a child to stay. (Every single placement is a city hall decision, not the social workers decision.)
There are some problems with them - lack of supervision.
http://www.secretprisonsforteens.dk/DK/Denmark.htmBut we do not have private programs, unless we are talking about religious sects and they are under constant attack from the authorities, because it is very costly for the taxpayers to undo brainwashing.
Are there adolescent psychiatric hospitals ?
Yes, we have them. They are runned by the state. We have a commitee from our parliaments who are visiting them unannounced due to the controversial nature of such facilities. Parents can not refer their child to the hospital themselves. They have to convince their family doctor to write a red or yellow note, so a special police unit can get the child to the hospital. Those hospitals are so costly due to our level of standard that the hospital would send the child out as fast as it came in, if there is no need for treatment.
We try our best in Denmark. Our biggest problem are lazy or indifferent parents, who are busy with their own life. Some seems to think that children are adults once they turn 12, so they skip supervising the important rite of passage where a child of 12 have exact 3-4 years to learn about sex and alcohol before they can buy it all on their own. (Regarding sex is legal, once they turn 15. It is not a crime in Denmark to buy sex. Some parents (I find it sick) have bought a prostitute for their son. Unfortunately rise in cost of education have lead some girls into semi-prostitution where they get books, talktime to their mobile phone, clothes etc. instead of money for sex.)
So a little more supervision would be good in our country. Because most things is not illegal (We have no limit of alcohol consumption), those issues should be relatively easy to discuss and there should be no reason for the child to keep it secret.