By Kocsis I.

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D´ avila, L. Dupaigne, Hardy-type inequalities, J. Eur. Math. Soc. (JEMS), 6 (2004), no. 3, 335–365. [10] V. De Cicco, M. A. Vivaldi, Harnack inequalities for Fuchsian type weighted elliptic equations, Comm. Partial Differential Equations, 21 (1996), no. 9-10, 1321–1347. [11] T. Duyckaerts, Op´ erateur de Schr¨ odinger avec potentiel singulier multipolaire, Bulletin de la SMF, to appear. [12] H. Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ.

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M ) < Scirc (λ0 , Λ1 , . . , Λm ) + ε. 3 and (107), there exists k2 such that for all k ≥ k2 (110) Scirc (λ0 , Λ1 , . . , Λm ) + ε < Sk (λ0 ) ≤ Scirc (λ0 ). Let k3 be such that for all k ≥ k3 (111) 2 2 2 Scirc (λ0 , Λ1 , . . , Λm ) + ε ≤ min k N S, k N S(λ1 ), . . , k N S(λm ) . From (108–111), we conclude that for all k ≥ max{k1 , k2 , k3 } 2 2 2 Sk (λ0 , Λ1 , . . , Λm ) < min k N S, k N S(λ1 ), . . , k N S(λm ), Sk (λ0 ), Sk λ0 + k m =1 λ . 1, we deduce that {un }n∈N has a subsequence strongly converging to some u0 ∈ Dk1,2 (RN ) such that Jk (u0 ) = N1 Sk (λ0 , Λ1 , .

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