Learn Turkish: Gözünde bir problem var. - There is a problem in his eye.

The root word is göz (eye). We add -ü to mean "his" or "her" eye. To say "in his eye", we need to add the locative suffix -de (in/at/on). In Turkish, whenever you add a case suffix (like -de) to a third-person possessive suffix (like -ü), you must insert the buffer letter -n- between them. So, it becomes göz + ü + n + de.

Wait, doesn't gözünde also mean "in your eye"?

Yes, it can! If you want to say "your eye", you add the 2nd person possessive -ün to get gözün, and then add -de to get gözünde. Because of the buffer -n- in the third-person version, "in his/her eye" and "in your eye" look identical. You usually tell them apart from the context of the conversation, or by adding onun (his/her) or senin (your) before the word.

Why does göz take the -ü suffix for "his" instead of -i?

This follows the rules of 4-way vowel harmony. Since the vowel in göz is ö, the possessive suffix must use ü. If the word had a front unrounded vowel like e or i (for example, diş meaning tooth), the suffix would be -i (making dişinde).