Affiliations: [a] School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844-13114, Iran. E-mails: [email protected], [email protected] | [b] School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box: 19395-5746, Tehran, Iran
Abstract: We consider the nonlinear eigenvalue problem Lu=λf(u), posed in a smooth bounded domain Ω⊆RN with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, λ>0 and f:[0,af)→R+ (0<af⩽∞) is a smooth, increasing and convex nonlinearity such that f(0)>0 and which blows up at af. First we present some upper and lower bounds for the extremal parameter λ∗ and the extremal solution u∗. Then we apply the results to the operator LA=−Δ+Ac(x) with A>0 and c(x) is a divergence-free flow in Ω. We show that, if ψA,Ω is the maximum of the solution ψA(x) of the equation LAu=1 in Ω with Dirichlet boundary condition, then for any incompressible flow c(x) we have, ψA,Ω⟶0 as A⟶∞ if and only if c(x) has no non-zero first integrals in H01(Ω). Also, taking c(x)=−xρ(|x|) where ρ is a smooth real function on [0,1] then c(x) is never divergence-free in unit ball B⊂RN, but our results completely determine the behaviour of the extremal parameter λA∗ as A⟶∞.
