Like with any other credit, you are obliged to repay the loan to the bank. If for some reason you do not honor your home loan repayments the bank has the right to demand you to sell the house. This is normally the best solution for both parties as with the uncertainty around the final price at auctions (with the biggest risk being that the sales price would be below the remaining loan and you are left with the remaining debt), it is neither in your interest nor that of the bank to proceed with a public auction. With the proceeds of the sale you can repay all outstanding amounts of the loan(s) plus due interests to the bank and keep the remaining money.
Most cases revolving around breaches in terms of mortgage loan repayments in Luxembourg are settled between the bank and the owner, ending either with adjustments in the terms of the home loan contract, if the issue is temporary or, if necessary with a normal sale of the property by the owner on the open market.