We will consider a SUSY-SU(5) with one right-handed neutrino with a large top like Yukawa coupling. Assuming universal soft masses at high scale we compute the low-energy spectrum and subsequently the neutralino LSP relic density taking also into consideration SU(5) as well as the see-saw running effects above the gauge coupling unification scale. We found that there exists no viable region in parameter space for $\tanβ\ler ~35$. The $\tildeτ$ coannihilation process starts becoming efficient for $\tanβ\ger 35-40$. However, this process is significantly constrained by the limited range in which the stau is lighter than the neutralino. In fact, for a given $\tanβ$ we find that there exists an upper bound on the lightest neutralino mass ($M_{χ_1^0}$) in this region. The A-pole funnel region appears at very large $\tanβ\simeq 45-50$, while the focus-point region does not make an appearance till large ($m_0,M_{1/2}$), namely a few TeV. Large $A_0$ terms at high scale can lead to extended regions consistent with WMAP constraints and remove the upper bounds in the stau coannihilation regions.

18 pages, 7 figures; references added, figure added, improved discussions in text, to appear in JHEP