Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems

Abstract

We study the nonlinear two-parameter problem $-u^{″}\left(x\right)+\lambda u{\left(x\right)}^q=\mu u{\left(x\right)}^p$, $u\left(x\right)>0$, $x\in \left(0,1\right)$, $u\left(0\right)=u\left(1\right)=0$. Here, $1<q<p$ are constants and $\lambda ,\mu >0$ are parameters. We establish precise asymptotic formulas with exact second term for variational eigencurve $\mu \left(\lambda \right)$ as $\lambda \to \infty$. We emphasize that the critical case concerning the decaying rate of the second term is $p=\left(3q-1\right)/2$ and this kind of criticality is new for two-parameter problems.