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

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Abstract

This paper is concerned with non-linear elliptic problems of the type (Pε): -Δu = u(N + 2) (N - 2) + εu, u > 0 on Ω; u = 0 on ∂Ω, where Ω is a smooth and bounded domain in RN, N ≥ 4, and ε > 0. We show that if the uε are solutions of (Pε) which concentrate around a point as ε → 0, then this point 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 x0 of the regular part of the Green's function, there exist solutions of (Pε) concentrating around x0 as ε → 0. © 1990.

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Rey, O. (1990). The role of the Green’s function in a non-linear elliptic equation involving the critical Sobolev exponent. Journal of Functional Analysis, 89(1), 1–52. https://doi.org/10.1016/0022-1236(90)90002-3

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