Der findes en løsning.

Here der is an existential or dummy subject, like English there in There is... or There exists....

It does not mean there as a place in this sentence.

So in Der findes en løsning, der is just there because Danish often uses it when introducing the existence of something.

No. In this sentence, der does not mean a place.

Compare:

• Der ligger bogen. = The book is lying there.
  
  • Here der is locational.
• Der findes en løsning. = There exists a solution.
  
  • Here der is only grammatical; it does not point to any place.

The learner question is very common because English also uses there in both ways.

Findes comes from the verb at finde, which usually means to find.

But in the expression der findes, it often means:

• exists
• can be found
• is available

So in this sentence, findes is not really finds in the normal sense. It is part of a common Danish way to say that something exists.

Because findes is the -s passive form of at finde.

• finder = finds / is finding
• findes = is found / exists

In Danish, the -s ending is one way to form the passive. But with der findes, the expression is so common that learners often just treat it as a set phrase meaning there is or there exists.

They are close, but not always identical in tone.

• Der er en løsning = There is a solution
• Der findes en løsning = There exists a solution / A solution can be found

Der findes can sound a little more formal, abstract, or emphatic. It focuses more on the fact that a solution exists.

In everyday speech, der er is often more common.

Because en løsning is indefinite: it means a solution, not the solution.

This sentence introduces the idea that some solution exists, without saying which one it is.

• en løsning = a solution
• løsningen = the solution

If you said Der findes løsningen, that would usually sound wrong in normal Danish, because existential sentences like this normally use an indefinite noun phrase.

Because løsning is a common-gender noun in Danish.

Danish nouns have two grammatical genders:

• en-words = common gender
• et-words = neuter

So:

• en løsning
• løsningen

This is something you generally have to learn together with the noun.

The normal order here is:

Der + verb + noun phrase

So:

• Der = dummy subject
• findes = verb
• en løsning = the thing that exists

Danish main clauses usually follow the verb-second pattern, and this sentence fits that pattern neatly: der takes the first position, and the verb comes right after it.

A natural negated version is:

Der findes ikke en løsning.

That means There is not a solution or No solution exists.

But very often Danish would prefer:

Der findes ingen løsning.
or
Der er ingen løsning.

These often sound more natural than using ikke + en.

A rough pronunciation guide is:

dær finn-es en LØS-ning

A few useful points:

• der sounds somewhat like English there, but with Danish pronunciation.
• findes has two syllables: fin-des
• løsning has the stress on the first syllable: LØS-ning

The Danish ø sound in løsning is important and may be unfamiliar to English speakers.

Yes. Der findes is a productive pattern in Danish.

For example:

• Der findes mange muligheder. = There are many possibilities.
• Der findes ingen nem løsning. = There is no easy solution.
• Der findes folk, som mener det modsatte. = There are people who think the opposite.

So once you understand der findes, you can reuse it in many similar sentences.

It is generally neutral to slightly formal.

It is perfectly normal Danish, but compared with Der er en løsning, it may sound a bit more:

• careful
• written
• abstract
• analytical

So a native speaker would understand it immediately, but in casual everyday conversation they might more often choose Der er en løsning.