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⟶∞.

中文翻译：

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非线性特征值问题的极值参数界线及应用

我们考虑非线性特征值问题Lu =λf（u），它存在于Dirichlet边界条件下的光滑有界域Ω⊆RN中，其中L是一个均匀椭圆的二阶线性微分算子，λ> 0且f：[0，af ）→R +（0 <af⩽∞）是光滑的，渐增的和凸的非线性，使得f（0）> 0并在af处爆炸。首先，我们给出了极值参数λ∗和极值解u ∗的上下界。然后，将结果应用于A> 0的算子LA =-Δ+ Ac（x），并且c（x）是以Ω为单位的无散度流。我们证明，如果ψA，Ω是Dirichlet边界条件下方程LAu = 1在Ω中的解ψA（x）的最大值，则对于任何不可压缩流c（x），我们的ψA，Ω⟶0为A当且仅当c（x）在H01（Ω）中没有非零的第一积分时，才为∞。另外，取c（x）=-xρ（| x |），其中ρ是[0，