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Article Dans Une Revue Journal of Functional Analysis Année : 1990

The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent

Olivier Rey

Résumé

This paper is concerned with nonlinear elliptic problems of the type (P_ε): -Δu = u^(N+2)/(N-2) + εu, u>0 in Ω; u=0 on ∂Ω, where Ω is a smooth and bounded domain in R^N, N≥4, and ε>0. We show that if the u_ε are solutions to (P_ε) which concentrate at a point x_0 as ε goes to 0, x_0 cannot be on the boundary of Ω and is a critical point of the regular part of the Green's function. Conversely, we show that for N≥5 and any non-degenerate critical point x_0 of the regular part of the Green's function, there exist solutions of (P_ε) concentrating at x_0 as ε goes to 0.
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Dates et versions

hal-00935374 , version 1 (12-01-2017)

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  • HAL Id : hal-00935374 , version 1

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Olivier Rey. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent. Journal of Functional Analysis, 1990, 89 (1), pp.1-52. ⟨hal-00935374⟩
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