Is there any written evidence in Mahabharata why Krishna supported Pandavas in war?
Yes, the Pandavas were incarnations of the Devas, whereas the Kauravas were incarnations of Rakshasas and Asuras, and the Mahabharata war was actually a war between the Devas and Asuras for supremacy of the Earth.
From the Adivansavatarana Parva:
"And, O bull of the Bharata race, when such was the blessed state of
the terrestrial world, the Asuras, O lord of men, began to be born in
kingly lines. And the sons of Diti (Daityas) being repeatedly defeated
in war by the sons of Aditi (celestials) and deprived also of
sovereignty and heaven, began to be incarnated on the earth. And, O
king, the Asuras being possessed of great powers, and desirous of
sovereignty began to be born on earth amongst various creatures ... And, O protector of the earth, owing to those already born and to those that were being born, the earth became incapable of supporting herself. And amongst the sons of Diti and of Danu, cast out of heaven, some were born on the earth as kings of great pride and insolence. Possessed of great energy, they covered the earth in various shapes. Capable of oppressing all foes, they filled the earth having the ocean for its boundaries. And by their strength they began to oppress Brahmanas and Kshatriyas and Vaisyas and Sudras and all other creatures also. Terrifying and killing all creatures, they traversed the earth, O king, in bands of hundreds and thousands. Devoid of truth and virtue, proud of their strength, and intoxicated with (the wine of) insolence, they even insulted the great Rishis in their hermitages.
Then, the Goddess of Earth, being overburdened with Asuras, went to Lord Brahma for help:
"And the earth, thus oppressed by the mighty Asuras endued with great strength and energy and possessed of abundant means, began to think of waiting on Brahman... And then, O king, the earth, oppressed with weight and afflicted with fear, sought the protection of the Grandsire of all creatures. And she beheld the divine Brahman--the Creator of the worlds who knoweth no deterioration
Brahma ordered the Devas to incarnate on Earth and fight the Asuras:
And the Creator then commanded all the gods saying, 'To ease the Earth of her burden, go ye and have your births in her according to your respective parts and seek ye strife (with the Asuras already born there)'. And the Creator of all, summoning also all the tribes of the Gandharvas and the Apsaras, spake unto them these words of deep import, 'Go ye and be born amongst men according to your respective parts in forms that ye like.'
Then, Indra went and asked Narayana to incarnate on Earth too to help them:
Him, Indra the most exalted of persons, addressed, saying, "Be incarnate." And Hari replied,--'Let it be.'"
And the dwellers in heaven gradually became incarnate on earth for the destruction of the Asuras and for the welfare of the three worlds.
The Vishnu Purana also says the same thing:
At this present season many demons, of whom Kálanemi is the chief, have overrun, and continually harrass, the region of mortals. The great Asura Kálanemi, that was killed by the powerful Vishńu, has revived in Kansa, the son of Ugrasena, and many other mighty demons, more than I can enumerate, as Arisht́a, Dhenuka, Keśin, Pralamba, Naraka, Sunda, and the fierce Báńa, the son of Bali, are born in the palaces of kings.
When Brahmá had ended, the supreme lord plucked off two hairs, one white and one black, and said to the gods, "These my hairs shall descend upon earth, and shall relieve her of the burden of her distress. Let all the gods also, in their own portions, go down to earth, and wage war with the haughty Asuras, who are there incorporate, and who shall every one of them be destroyed.
So, it only makes sense that Krishna, who is an incarnation of Vishnu, supported the Pandavas.