Bounds for the extremal parameter of nonlinear eigenvalue problems and applications

Abstract

We consider the nonlinear eigenvalue problem L u = λ f ( u ), posed in a smooth bounded domain Ω ⊆ R N with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, λ > 0 and f : [ 0 , a f ) → R + ( 0 < a f ⩽ ∞) is a smooth, increasing and convex nonlinearity such that f ( 0 ) > 0 and which blows up at a f . 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 L A = − Δ + A c ( x ) with A > 0 and c ( x ) is a divergence-free flow in Ω. We show that, if ψ A , Ω is the maximum of the solution ψ A ( x ) of the equation L A u = 1 in Ω with Dirichlet boundary condition, then for any incompressible flow c ( x ) we have, ψ A , Ω ⟶ 0 as A ⟶ ∞ if and only if c ( x ) has no non-zero first integrals in H 0 1 ( Ω ). Also, taking c ( x ) = − x ρ ( | x | ) where ρ is a smooth real function on [ 0 , 1 ] then c ( x ) is never divergence-free in unit ball B ⊂ R N , but our results completely determine the behaviour of the extremal parameter λ A ∗ as A ⟶ ∞.