We examine the elliptic system given by
where λ,γ are positive parameters, Ω is a smooth bounded
domain in \IR^{N} and f is a C^{2} positive, nondecreasing and
convex function in [0,∞) such that [(f(t))/(t)]→∞ as t→∞. Assuming
0 < τ_{−}:= 
liminf
t→∞


f(t)f"(t)
f′(t)^{2}

≤ τ_{+}:= 
limsup
t→∞


f(t)f"(t)
f′(t)^{2}

≤ 2, 

we show that the extremal solution (u^{*}, v^{*}) associated to the above system is smooth provided that N < [(2α_{*}(2−τ_{+})+2τ_{+})/(τ_{+})]max{1,τ_{+}}, where α_{*} > 1 denotes the largest root of the second order polynomial
P_{f}(α,τ_{−},τ_{+}):=(2−τ_{−})^{2} α^{2}− 4(2−τ_{+})α+4(1−τ_{+}). 

As a consequence, u^{*}, v^{*} ∈ L^{∞}(Ω) for N < 5. Moreover, if τ_{−}=τ_{+}, then u^{*}, v^{*} ∈ L^{∞}(Ω) for N < 10.
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