We establish the existence of nodal solutions for the problem -ε~2Δu + u = |u|~(p-2)u in Ω, u ∈ H_0~1(Ω), where Ω is a bounded domain, 2 < p < 2N/(N - 2) for N ≥ 3, and 2 < p < ∞ for N = 2. It is shown that, corresponding to each pair of points P_1, P_2 ∈ (Ω * Ω), whose location depends on the geometry of Ω, there is a nodal solution, which has, for small ε, exactly one positive and one negative peak, and that the peak points converge, as ε → 0, to (P_1, P_2).
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机译：我们建立了问题-ε〜2Δu+ u = | u |〜（p-2）u在Ω中的节点解的存在，u∈H_0〜1（Ω），其中Ω是有界域，2 <对于N≥3，为2N /（N-2），对于N = 2，为2 <∞。表明，对应于每对点P_1，P_2∈（Ω*Ω），其位置取决于几何形状对于Ω，存在一个节点解，对于小ε，正好有一个正峰和一个负峰，并且峰点从ε→0收敛到（P_1，P_2）。
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