sinkokuna mondai ga arimasu.

深刻な is a な-adjective (also called a “na-adjective”). Unlike い-adjectives (which end in い, like 大きい), な-adjectives:

• Behave more like nouns when they modify another noun.
• Require the particle な to connect to the noun that follows.
  So you attach な to 深刻 and then add 問題, giving 深刻な問題 (“a serious problem”).

Key differences:

• Ending:
  
  • い-adjectives end with い (e.g. 暑い, 楽しい).
  • な-adjectives do not end with い in their base form (e.g. 静か, 複雑, 深刻).
• Noun modification:
  
  • い-adjective
    • noun: 大きい家
  • な-adjective
    • な + noun: 静かな部屋
• Adverbial form:
  
  • い-adj. → replace い with く: 早く歩く
  • な-adj. → add に: 静かに話す
• Conjugation:
  
  • Past tense:
    
    • い-adj.: 暑い → 暑かった
    • な-adj.: 静か → 静かだった

Japanese distinguishes animate vs. inanimate existence:

• います is used for animate beings (people, animals).
• あります is used for inanimate things or abstract concepts (objects, events, problems).
  Since 問題 (problem) is abstract/inanimate, we use あります.

Yes. Any noun you want to express as “existing” can follow this pattern. Examples:

• 大きな誤りがあります。 (“There is a big mistake.”)
• 面白い質問があります。 (“I have an interesting question.”)
• 新しい情報があります。 (“There is new information.”)