The effect of the variational framework on the spectral asymptotics for two–parameter nonlinear eigenvalue problems

We consider the nonlinear two–parameter problem u ″( x ) + μu ( x ) p = λu ( x ) p , u ( x ) > 0, x ∈ I = (0, 1), u (0) = u (1) = 0, where 1 < q < p < 2 q + 3 and λ , μ > 0 are parameters. We establish the three–term spectral asymptotics for the eigencurve λ = λ ( μ ) as μ → ∞ by using a variational method on the general level set due to Zeidler. The first and second terms of λ ( μ ) do not depend on the relationship between p and q . However, the third term depends on the relationship between p and q , and the critical case is p = (3 q – 1)/2.