On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
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DOI：
10.1017/s0308210500013147
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发表时间：
1999
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影响因子：
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通讯作者：
L. Jeanjean
中科院分区：
文献类型：
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作者：
L. Jeanjean
利用Struwe引入的“单调性技巧”，我们得到了一个一般定理。它指出，对于一大类具有山路(MP)几何的泛函，这类泛函中的几乎每个泛函在MP级都有一个有界的Palais-Smear序列。然后，我们展示了对于给定的泛函，如何利用这个一般定理来得到一个特殊的Palais-Smer序列，该序列具有额外的性质，有助于确保它的收敛。随后，这些抽象结果被应用于证明如下形式的问题的正解的存在性：我们假设与(P)相关的泛函具有MP几何。我们的结果涵盖了如下情形：f满足(I)f(x，S)S−1→a∈(0，∞)作为S→+∞；和(Ii)f(x，S)S-1作为S≥0，A.E.的函数非递减。X→ℝN.
Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.