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Paper
IPM / M / 15178 |
School of Mathematics
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Title: |
Regularity of the extremal solutions associated to elliptic systems
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Author(s): |
Asadollah Aghajani (Joint with C. Cowan)
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Status: |
To Appear
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Journal: |
Proc. Edinburgh Math. Soc.
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Supported by: |
IPM
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Abstract: |
We examine the elliptic system given by
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
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0 < τ−:= |
liminf
t→∞
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f(t)f"(t)
f′(t)2
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≤ τ+:= |
limsup
t→∞
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f(t)f"(t)
f′(t)2
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≤ 2, |
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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
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Pf(α,τ−,τ+):=(2−τ−)2 α2− 4(2−τ+)α+4(1−τ+). |
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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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