Sun is considered king among the planets in vedic astrology. A weak Sun is certainly undesirable in a horoscope. Below are the points where Sun can be considered weak.

1. If Sun is weak than a native lacks the necessary will-power to take a firm stand on his principles 
2. Sun is the significator of health and if health of native remains very weak than he could have a weak Sun. Bear in mind First house and its lord, Sixth House and Eight House also play important role in deciding health of native. It isn't always Sun's fault for weak health.
3. If the native suffers from frequent boils, burns, haemorrhoids(piles), pimples and dehydration than it could be due to weak Sun or Sun in problematic location.
4. If there are constant raids or harassment from government and tax-department then also it could be possible that Sun is in a dushamsthanas or with malefics.
5. A weak Sun ensures that no accolades are derived by native for his/her efforts. Also if native works in a high-end job but lacks the power and prestige associated with similar jobs than also there is a possibility of weak Sun in horoscope. For example software professionals and diplomats work behind the scenes and their work does not get the respect it deserves.
6. In a horoscope Sun's association with Rahu or Saturn via placement or aspect is a sign of weak Sun in most cases. In rare cases it does give good results.

     Natives with weak Sun should regularly chant the Gayatri Mantra. Also they should pray to Lord Vishnu. Vishnu is the greatest among the Twelve Adityas that includes Sun. Regularly chanting the Vishnu Sahasranaam would do wonders for natives with weak Sun. But ideally the simple Gayatri Mantra would suffice. It is the greatest mantra in Vedas and nothing else is needed once you speak it. There is Surya Japa Mantra by great sage Ved Vyas for instant remedy if required.

Surya Japa Mantra

जपाकुसुम संकाशं काश्यपेयं महदद्युतिम्|
तमोरिंसर्वपापघ्नं प्रणतोSस्मि दिवाकरम्||

Japa Kusuma Samkaasham Kaashya-peyam Mahaa-dyutim
Tamorim Sarva-papaghnam Pranatosmin Divakaram

Surya Gayatri Mantra

Om Bhaskaraya Vidhmahe Diva karaya Dheemahe Thanno Surya Prachodayath.

     Twelve Poses of Surya Namaskar are ideal to aid problems related to weak Sun. They aid digestion, improve skin radiance, remove skin problems, help burn excessive fat, remove constipation, give energy to the body. Vrikshasana along with pranayam Anulom Vilom is helpful to improve concentration and create a balanced mind.

     Natives with weak Sun should donate pulses and clothes to the poor and needy. Keep a clean balance sheet of payments and taxes. Don't try to be over smart with government officials and Income Tax (IRS). Give due respect to the government officials and employers. Appreciate the efforts and hard work of your employees and co-workers.

     A weak Sun is often a result of having taken more credit than worthy in previous lives. Also having been too arrogant and discarding the efforts and feelings of others. The current life is a leveler so give the respect others deserve, be humble and try to be gracious in victory and defeat. Sooner or later your Sun would improve.