Allah's Name Al-Qayyum as Proof of His Existence
Question

Does Al-Qayyum mean The Self Existing One? Does it come from قَآئِمٌ which means Existing? If not, so is there any name of Allah which means Existing?  is مَوْجُود an urdu, persian or arabic word! If it is an arabic word so why it is not mentioned in Quran rather وَجَدَ is mentioned in Quran which means \"Found\" but مَوْجُود is not mentioned?
Praise be to Allah.Firstly:
The Name of Allah, Al-Qayyum, appears in verses of the Quran, including Allah's statement (translation of the meaning): {Allah - there is no deity except Him, the Ever-Living, the Self-Sustaining. Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them, and they encompass not a thing of His knowledge except for what He wills. His Kursi extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great.} (Al-Baqarah: 255), and His statement (translation of the meaning): {Alif, Lam, Mim. Allāh - there is no deity except Him, the Ever-Living, the Self-Sustaining.} (Al `Imran: 1-2), and His statement (translation of the meaning): {And [all] faces will be humbled before the Ever-Living, the Self-Sustaining. And he will have failed who carries injustice.} (Taha: 111).
Allah has combined in these verses two of His names, Exalted and Praised be He, which are Al-Hayy (the Ever-Living) and Al-Qayyum (the Sustainer of [all] existence), "for they encompass all the attributes of perfection. Al-Hayy is the One with perfect life, which includes all the attributes of Allah's Essence: such as Knowledge, Might, Power, Will, Greatness, Majesty, and other sacred attributes of His Essence.
Al-Qayyum is the One with perfect self-subsistence and it has two meanings:
The first meaning: He is the One who is self-subsisting, His attributes are magnificent, and He is independent of all His creation.
The second meaning: He is the One by Whom the earth and the heavens and all their creatures subsist. He is the One Who brought them into existence, sustains them, and prepared them for everything that ensures their survival, rectitude, and stability.
He is rich beyond need of them in every way, and they are the ones in need of Him in every way. The Ever-Living and the Sustainer of [all] existence is the One Who possesses the attribute of every perfection and He is the Doer of whatever He wills," End quote.
Sharh Asma' Allah Al-Husna Fi Daw' Al-Kitab was As-Sunnah": (159)".
And the Name of Allah, Al-Qayyum, also implies the existence of Allah, indeed, all His Exalted names carry this meaning. Al-Wahidi says: "And His (Exalted be He) statement {Al-Qayyum} Al-Qayyum linguistically is an intensive form of Al-Qa'im (the One who stands/exists), and its pattern is Fa`yul, and its root is: Qaywum, but when the 'ya' and 'waw' came together, and the preceding is a sukun, they were made into a shaddah of 'ya', and it is not permissible to be on (Fa`ul), because if it were so, it would have been Qawwman.
It has three dialects: Qayum, Qayyam, and Qayyim.
... As for its meaning: Mujahid said: Al-Qayyum: the One Who stands (exists) over everything, and its interpretation: that He stands by managing the affairs of creation, in their creation and their sustenance. And Ad-Dahhak said: Al-Qayyum: the One Whose existence is perpetual.
Abu `Ubaidah: He is the One Who does not cease. Due to the uprightness of His attribute of existence, where change is not permissible in any way.
And it is said: It means the Knower of affairs, from their saying: So-and-So stands by this book, meaning: He is knowledgeable about it," End quote.
At-Tafsir Al-Basit" (4/ 347 - 348)".
At-Tibi said: "(Al-Qayyum) .. its meaning is the One Who stands (exists) by Himself and sustains others, and this is absolutely and generally only correct for Allah, the Exalted, for His self-subsistence does not depend in any way on others, and the subsistence of everything is by Him; as the existence and continuity of things cannot be conceived except by His existence," End quote.
""Sharh Al-Mishkah" by At-Tibi: (6/ 1803).
Among those who stated that the name of Allah, Al-Qayyum, conveys the meaning of existence is "Ibn Abi Al-`Izz Al-Hanafi," who said (may Allah have mercy on him): "And from it: that He is Al-Qayyum Who does not sleep, as He is uniquely free from sleep and slumber, unlike His creation, for they sleep. And in this is an indication that the negation of anthropomorphism is not meant to negate the attributes, but rather He, Exalted be He, is described with the attributes of perfection, due to the perfection of His Essence.
So, the Ever-Living with an everlasting life does not resemble the living with a transient life, and for this reason, the worldly life is enjoyment, amusement, and play, and indeed, the Hereafter is the true life. Thus, the worldly life is like sleep, and the life of the Hereafter is like wakefulness.
It is not said: So this life of the Hereafter is perfect, and it is for the creature: because we say: The Ever-Living Whose life is from the attributes of His Essence that are inherent to it, He is the One Who granted the creature that perpetual life, so it is lasting by Allah's making it last, not that permanence is an inherent attribute of it by itself, unlike the life of the Lord, Exalted is He. And likewise, all His other attributes, so the attributes of the Creator are as befits Him, and the attributes of the creature are as befits it.
Know that these two names, I mean: the Ever-Living, the Sustainer of [all] existence, are mentioned together in the Quran in three Surahs as previously mentioned, and they are among the greatest names of Allah, to the extent that it was said: They are the Greatest name, for they encompass the affirmation of the attributes of perfection in the most complete and truthful manner, and Al-Qayyum indicates the meaning of Eternality and Everlastingness that the term Al-Qadim (the Ancient) does not indicate.
And it also indicates His existence by Himself, which is the meaning of His being necessarily existent.
Al-Qayyum is more eloquent than Al-Qayyam because the 'waw' is stronger than the 'alif', and it implies His standing/existing by Himself, by the agreement of the interpreters and the linguists, and it is known by necessity. And does it imply His sustaining others and His standing over them? There are two opinions, the most correct of which is: that it does imply that. And it implies the permanence of His standing/existence and the perfection of His standing/existence, due to the intensification in it, for He, Exalted be He, does not fade nor does He wane, for the fading one has certainly ceased, meaning: He does not disappear, nor diminish, nor perish, nor become non-existent, rather He is the Everlasting, the Remaining, who has always been and will always be, described with the attributes of perfection. The conjunction with the Ever-Living necessitates all other attributes of perfection, and indicates their permanence and continuity, and the absence of deficiency and non-existence from them eternally and perpetually. And for this reason, His statement (translation of the meaning): {Allah - there is no deity except Him, the Ever-Living, the Self-Sustaining.} [Al-Baqarah: 255] is the greatest verse in the Quran, as it has been established in the authentic [narrations] from the Prophet (peace and blessings be upon him)," End quote.
"Shark At-Tahawiyyah" (1/ 90 - 91).
Secondly:
It has been mentioned in several answers that it is permissible to describe Allah, the Exalted, as (Existing), and that the existence of Allah is known from the religion by necessity, and it is an attribute of Allah by the consensus of the Muslims, so it is reported about Him and (He is) described with it, but it is not one of His Beautiful Names (Al-Asma' Al-Husna).
Shaykh Al-Islam (may Allah have mercy on him) said: "And He, Exalted be He, is Ancient and  necessarily existent, the Lord of everything and its Sovereign, He is the Creator and all else is created," End quote.
"Clarification of the Confusion of the Jahmiyyah" (1/304).
He also said: "For Allah truly exists, and the servant truly exists, and this is not like that," End quote.
"Majmu` Al-Fatawa" (5/198).
It is a word and meaning in Arabic, as is known from the books of language, and although the phrase "Allah Exists" is not from the speech of the ancient Arabs, nor from their known poetry or prose; but it is common on the tongues of the scholars.
It is used in the context of Allah with two meanings: affirming Him, Exalted be He, and negating His non-existence, eternally and perpetually.
And affirming knowledge of Him also, so whoever seeks Him, finds Him, and knows Him, Exalted be He.
And Allah knows best.