Morse index and critical/super-critical Dirichlet problems with a large parameter

We consider the Dirichlet problem equation(1){−Δu+λu=upin Ωu=0on ∂Ω, with Ω⊂RNΩ⊂RN, N≥3N≥3, a bounded domain, with p≥N+2N−2, in the regime λ→+∞λ→+∞. In this paper, we study the asymptotic behavior of positive solutions of (1) and we obtain that solutions of (1) cannot have uniformly bounded Morse index as λ→+∞λ→+∞ as long as p