We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial. We also prove some intermediate results of independent interest on the localization and homotopy invariance of critical groups of functionals involving critical Sobolev exponents.

中文翻译：

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凹-线性-凸型椭圆问题的非平凡解对

我们获得了一类凹-线性-凸型椭圆问题的非平凡解，这些问题要么是临界的，要么是亚临界的。我们找到的解决方案既不是局部最小化器，也不是一般的山路类型。它们是更高的临界点，因为它们每个都有一个非平凡的更高临界组。这一事实对于证明我们的解决方案是非平凡的至关重要。我们还证明了一些对涉及关键 Sobolev 指数的关键泛函群的定位和同伦不变性具有独立兴趣的中间结果。