2017/07/21 by A. Aghajani, Aghajani, A., C. Cowan +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1707.06723
arxiv created 2017/07/21 · arxiv updated 2017/07/24
We examine the elliptic system given by \ -Δu =λf(v) in Ω -Δv =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-:=\liminft→∞ \fracf(t)f"(t)f'(t)2≤ τ+:=\limsupt→∞ \fracf(t)f"(t)f'(t)2≤ 2, we show that the extremal solution (u^*, v^*) associated to the above system is smooth provided N<\frac2α*(2-τ+)+2τ+τ+max\1,τ+\, where α*>1 denotes the largest root of the 2nd order polynomial Pf(α,τ-,τ+):=(2-τ-)2 α2- 4(2-τ+)α+4(1-τ+). As a consequences, u^*, v^*∈ L^∞(Ω) for N<5. Moreover, if τ-=τ+, then u^*, v^*∈ L^∞(Ω) for N<10.