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Global multiplicity for very-singular elliptic problems with vanishing non-local terms
Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2021-06-16 , DOI: 10.1007/s00033-021-01573-z
Willian Cintra , Carlos Alberto Santos , Lais Santos

In this paper, we deal with issues related to global multiplicity of \(W^{1,p}_{\mathrm {loc}}(\Omega )\)-solutions for the very-singular and non-local \(\mu \)-problem

$$\begin{aligned} -{g\left( \int \limits _\Omega u^q\right) }\Delta _pu={\mu } u^{-\delta } + u^{\beta } \ \ \text{ in } \ \ \Omega , \ \ \ \ u > 0 \ \ \ \text{ in } \ \Omega \ \ \ \text{ and } \ \ u=0 \ \ \text{ on } \ \partial \Omega , \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N \) is a smooth bounded domain, \(\delta >0\), \(q>0\), \(0 < \beta \le p-1\) and \(g{:}\,[0,\infty ) \rightarrow [0,\infty )\) is a continuous function that achieves critical values for the class of non-local problems (i.e., the level zero if \(\beta < p-1\) and \(1/\lambda _1\) if \(\beta = p-1\), where \(\lambda _1\) stands for the principal eigenvalue of the p-Laplacian in \(\Omega \) under homogeneous Dirichlet boundary conditions). To overcome the difficulties arising from the geometry of g and the presence of very-singular term combined with a \((p-1)\)-sublinear/asymptotically linear ones, we take advantage of a comparison principle for sub-supersolutions in \(W^{1,p}_{\mathrm {loc}}(\Omega )\)-sense proved in Santos and Santos (Z Angew Math Phys 69:Art. 145, 2018), together with sub-supersolutions techniques and bifurcation theory.



中文翻译:

具有消失非局部项的非常奇异椭圆问题的全局多重性

在本文中,我们处理与\(W^{1,p}_{\mathrm {loc}}(\Omega )\) 的全局多重性相关的问题 - 非常奇异和非局部\(\ mu \) - 问题

$$\begin{aligned} -{g\left( \int \limits _\Omega u^q\right) }\Delta _pu={\mu } u^{-\delta } + u^{\beta } \ \ \text{ in } \ \ \ u > 0 \ \ \ \ text{ in } \ \Omega \ \ \ \ text{ and } \ \ u=0 \ \ \ \ text{ on } \ \partial \Omega , \end{aligned}$$

其中\(\Omega \subset {\mathbb {R}}^N \)是一个光滑有界域,\(\delta >0\) , \(q>0\) , \(0 < \beta \le p -1\)\(g{:}\,[0,\infty ) \rightarrow [0,\infty )\)是一个连续函数,它实现了非局部问题类的临界值(即水平零如果\(\beta < p-1\)\(1/\lambda _1\)如果\(\beta = p-1\),其中\(\lambda _1\)代表p的主要特征值-在齐次狄利克雷边界条件下\(\Omega \) 中的拉普拉斯算子)。克服g的几何结构带来的困难并且非常奇异项的存在与\((p-1)\) -次线性/渐近线性项相结合,我们利用\(W^{1,p}_{ \mathrm {loc}}(\Omega )\) - 在 Santos 和 Santos (Z Angew Math Phys 69:Art. 145, 2018) 中证明的意义,以及子超解技术和分叉理论。

更新日期:2021-06-17
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