In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let X be a real Hilbert space. For each r > 0 , let S r = { x ∈ X : ∥ x ∥ 2 = r } . Let J : X → R be a sequentially weakly upper semicontinuous functional which is Gateaux differentiable in X ? { 0 } . Assume that lim?sup x → 0 J ( x ) ∥ x ∥ 2 = + ∞ . Then, for each ρ > 0 , there exists an open interval I ? ] 0 , + ∞ [ and an increasing function φ : I → ] 0 , ρ [ such that, for each λ ∈ I , one has ? ≠ { x ∈ S φ ( λ ) : J ( x ) = sup S φ ( λ ) J } ? { x ∈ X : x = λ J ′ ( x ) } .
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