In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Holder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering several different interactions between the nonlocal term and the nonlinearity.

中文翻译：

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退化基尔霍夫类型问题中的 Sobolev 与 Hölder 局部极小值

在本文中，我们研究了与具有基尔霍夫型退化非局部项的拟线性椭圆边界值问题相关的某些泛函的几何形状。由于非局部项的退化，不可能直接使用经典结果，例如统一先验估计和“Sobolev 与 Holder 局部最小化”类型的结果。我们证明与这些类似的结果是否成立，取决于问题的退化程度。考虑到非局部项和非线性之间的几种不同相互作用，我们应用我们的发现来显示相关拟线性方程的解的存在性和多重性。