We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208–233, 2005) to study the structure of positive solutions to the equation εm Δmu − um−1 + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of ℝN (N ≥ 2). First, we study subcritical case for 2 < m < N and show that after passing by a sequence positive solutions go to a constant in C1, α sense as ε → ∞. Second, we study the critical case for 1 < m < N and prove that there is a uniform upper bound independent of ε ∈ [1, ∞) for the least-energy solutions. Third, we show that in the critical case for 1 < m ≤ 2 the least energy solutions must be a constant if ε is sufficiently large and for 2 < m < N the least energy solutions go to a constant in C1, α sense as ε → ∞.
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机译:我们继续我们的工作(Y. Li,C. Zhao在J Differ Equ 212:208–233,2005中),研究方程ε m sup>Δmu-u m的正解的结构1 N sup>(N≥2)的光滑有界域中的−1 sup> + f(u)= 0,具有齐次Neumann边界条件。首先,我们研究2 1时变为一个常数ε→∞。其次,我们研究1 C 1,α sup> em>的含义为ε em>→∞。
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