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Given ( M , g ) ${(M,g)}$ , a smooth compact Riemannian n-manifold, we prove that for a generic Riemannian metric g the critical points of the function 𝒲 g ⁢ ( ξ ) := | Weyl g ⁢ ( ξ ) | g 2 ${\mathcal{W}_{g}(\xi):=\lvert\mathrm{Weyl}_{g}(\xi)\rvert^{2}_{g}}$ with 𝒲 g ⁢ ( ξ ) ≠ 0 ${\mathcal{W}_{g}(\xi)\not=0}$ are nondegenerate.

Given (M, g) the set {(ε, h) ∈ (0, 1) × Sk : any solution u ∈ A of -ε2 Δg+h u+u = (u+)p-1 on M is non degenerate} is an open dense subset of (0, 1) × Bρ where Bρ is a suitable ball centered at 0 with radius ρ in the Banach space Sk. Here Sk is the space of all Ck symmetric covariant 2-tensors on M and A is a bounded subset of Hg 1 (M) which does not contain the constant function 1.

We study an eigenvalue problem for functions in ℝN and we find sufficient conditions for the existence of the fundamental eigenvalue. This result can be applied to the study of the orbital stability of the standing waves of the nonlinear Schrödinger equation.

In this paper, we are concerned with the existence of solutions of the problem: where Ω ⊂ ℝN is an exterior domain and f″(0) = 0.