I write today with hope to you to find an answer to a question whichworries me for a long time: I am reached stammering since my youthand this handicap causes me hardship (I only speak veryseldom by fear of the mocking remarks, the prospect to find work withthis illness discourages me). To this end, is there a remedy in Islamwhich will enable me to completely overcome (or even partially) thestammering? I'm waiting your response impatiently and thank you...

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
The solution to your problem lies in asking Allah to cure you from this defect. Musa, the Prophet of Allah had a "Knot" in his tongue which made it difficult for him to make people understand what he said. When Allah Sent him to Pharaoh he said: "[Mûsa (Moses)] said: "O my Lord! Open for me my chest (grant me self-confidence, contentment, and boldness). "And ease my task for me; "And make loose the knot (the defect) from my tongue, (i.e. remove the incorrectness from my speech) [That occurred as a result of a brand of fire which Mûsa (Moses) put in his mouth when he was an infant]." [20:25-27].
Allah Answered his supplication and said to him: "Allâh said: "You are granted your request, O Mûsa (Moses)!"[20:36].
You should seek help from medical professionals for your condition, a specialized doctor. You should follow their advice if it does not contradict Sharia. If there is no change, then we remind you that you must accept the destiny of Allah and your fate.
Meanwhile, talk to people and try to get a job, as there are many jobs that do not need much talking.
Do not be discouraged that some people may laugh at you. Not everyone will. Later they may get used to your stammering, and accept you as you are. So, break this barrier between you and people and things will get much better, Insha Allah.
We ask Allah to ease your concerns and cure you as He cured His Prophet Musa.
Allah knows best.