Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena

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文摘

We exhibit a new concentration phenomenon for the problem

−ε²Δu+u=|u|^p−2u in Ω, u=0 on ∂Ω,

- ε^{2} Δ u + u = {| u |}^{p - 2} u in Ω, u = 0 on \partial Ω,

where Ω is some bounded smooth domain in R^N

R^{N}

, N≥4

N \geq 4

, ε>0

ε > 0

, and

p \in (2, \frac{2 N}{N - 2})

. We show that there is a sequence of sign changing solutions, which concentrate at a single point, whose asymptotic profile as ε→0

ε \to 0

is a rescaling of a nonradial sign changing solution to the limit problem

−Δu+u=|u|^p−2u, u∈H¹(R^N),

- Δ u + u = {| u |}^{p - 2} u, u \in H^{1} (R^{N}),

with sufficiently small energy.

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