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There exists ρ0 > 0 small such that the operator Sρ − Tρ is invertible with uniformly bounded inverse for ρ ∈ (0, ρ 0 ). Proof Since Sρ − Tρ → S0 − T0 as ρ → 0 in the operatorial norm, we want to prove that S0 − T0 is invertible. By an idea of R. Mazzeo used in the paper [4], we claim that it is enough to prove that S0 − T0 is injective. Regarding S0 − T0 as an operator from H 1 (∂B) into L2 (∂B), it is a self adjoint first order pseudodifferential operator. Since S 0 and T0 are elliptic with principal symbols −|ξ| and |ξ| respectively, the difference S 0 − T0 is also elliptic and semibounded.

46(1993), pp. 27-56. [24] J. Liouville, Sur l’´equation aud d´eriv´ees partielles ∂ 2 log λ/∂u∂v ±2λa2 = 0, J. , 18(1853), pp. 71-72. [25] F. Mignot, F. Murat, J. P. Puel, Variation d’un point de retournement par rapport au domaine, Comm. Partial Differential Equations, 4(1979), pp. 1263-1297. [26] J. L. Moseley, A two-dimensional Dirichlet problem with an exponential nonlinearity, SIAM J. Math. , 14(1983), pp. 934-946. [27] K. Nagasaki, T. , 3(1990), pp. 173188. [28] W. M. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm.

25] F. Mignot, F. Murat, J. P. Puel, Variation d’un point de retournement par rapport au domaine, Comm. Partial Differential Equations, 4(1979), pp. 1263-1297. [26] J. L. Moseley, A two-dimensional Dirichlet problem with an exponential nonlinearity, SIAM J. Math. , 14(1983), pp. 934-946. [27] K. Nagasaki, T. , 3(1990), pp. 173188. [28] W. M. Ni, J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. , 48(1995), pp. 731768.

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