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Paper   IPM / M / 15178
 School of Mathematics Title: Regularity of the extremal solutions associated to elliptic systems Author(s): Asadollah Aghajani (Joint with C. Cowan) Status: To Appear Journal: Proc. Edinburgh Math. Soc. Supported by: IPM
Abstract:
We examine the elliptic system given by

 −∆u = λf(v)     in Ω,
 −∆v = γf(u)     in Ω,
 u=v = 0     on \pOm,
where λ,γ are positive parameters, Ω is a smooth bounded domain in \IRN and f is a C2 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
 Pf(α,τ−,τ+):=(2−τ−)2 α2− 4(2−τ+)α+4(1−τ+).
As a consequence, u*, v*L(Ω) for N < 5. Moreover, if τ+, then u*, v*L(Ω) for N < 10.