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Positive solutions of superlinear indefinite prescribed mean curvature problems
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2020-07-27 , DOI: 10.1142/s0219199720500170
Pierpaolo Omari 1 , Elisa Sovrano 2
Affiliation  

This paper analyzes the superlinear indefinite prescribed mean curvature problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with a regular boundary [Formula: see text], [Formula: see text] satisfies [Formula: see text], as [Formula: see text], [Formula: see text] being an exponent with [Formula: see text] if [Formula: see text], [Formula: see text] represents a parameter, and [Formula: see text] is a sign-changing function. The main result establishes the existence of positive regular solutions when [Formula: see text] is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for [Formula: see text] small is further discussed assuming that [Formula: see text] satisfies [Formula: see text] as [Formula: see text], [Formula: see text] being such that [Formula: see text] if [Formula: see text]; thus, in dimension [Formula: see text], the function [Formula: see text] is not superlinear at [Formula: see text], although its potential [Formula: see text] is. Imposing such different degrees of homogeneity of [Formula: see text] at [Formula: see text] and at [Formula: see text] is dictated by the specific features of the mean curvature operator.

中文翻译:

超线性不定规定平均曲率问题的正解

本文分析了超线性不定规定平均曲率问题[公式:见文] 其中[公式:见文]是[公式:见文]中的有界域,有正则边界[公式:见文],[公式:见文] ] 满足 [Formula: see text], as [Formula: see text], [Formula: see text] 是与 [Formula: see text] 的指数 if [Formula: see text], [Formula: see text] 表示参数, 而 [公式:见正文] 是一个符号变换函数。当[公式:见文本]足够大时,主要结果确定了正规则解的存在,并提供了有关解集结构的一些信息。[公式:见文本] small 的正有界变化解的存在性进一步讨论假设 [公式:见文本] 满足 [公式:见文本] 为 [公式:见文本],[公式:见正文] 使得 [公式:见正文] 如果 [公式:见正文];因此,在维度 [Formula: see text] 中,函数 [Formula: see text] 在 [Formula: see text] 处不是超线性的,尽管它的势 [Formula: see text] 是超线性的。在 [公式:参见文本] 和 [公式:参见文本] 处施加 [公式:参见文本] 的这种不同程度的同质性是由平均曲率算子的特定特征决定的。
更新日期:2020-07-27
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