Asymptotic formulas and critical exponents for two-parameter nonlinear eigenvalue problems (Q1811855)

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