Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems (Q2569844)

The author considers the two-parameter nonlinear eigenvalue problem \[ -\Delta u= \mu u -\lambda (u+u^p +f(u)), \quad u>0\,\,\text{in}\,\, \Omega, \quad u=0 \,\,\text{on}\,\, \partial\Omega. \] The author establishes asymptotic formulas for the variational eigencurves \(\lambda =\lambda(\mu, \alpha)\) as \(\mu\rightarrow\infty\), and emphasizes that the critical case from the viewpoint of two-term asymptotics of the eigencurve is \(p=3\). Moreover, it is shown that \(p=5/3\) is also a critical exponent from the viewpoint of three-term asymptotics when \( \Omega\) is a ball or an annulus.