ABSTRACT

In this paper, we consider a nonlinear fractional equation with critical exponent on higher dimensional bounded domain. Due to the existence of critical points at infinity, this problem present a lack of compactness. To overcome this difficulty, we prove a version of Morse lemma at infinity for this problem. Then we characterize the critical points at infinity associated to this problem which allows us to prove a new existence result. Moreover, for generic function K, we give a lower bound for the number of solutions in terms of the topological contribution of the critical points at infinity.

中文翻译：

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关于高维域上的非线性椭圆分数方程

摘要

在本文中，我们考虑了一个在高维有界域上具有临界指数的非线性分数方程。由于无穷处存在临界点，这个问题缺乏紧致性。为了克服这个困难，我们证明了这个问题的无穷远处莫尔斯引理的一个版本。然后我们描述了与这个问题相关的无穷远临界点，这使我们能够证明一个新的存在结果。此外，对于泛型函数K，我们根据无穷大临界点的拓扑贡献给出解数的下限。