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

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.

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

其中\(\Omega \subset {\mathbb {R}}^N \)是一个光滑有界域，\(\delta >0\) , \(q>0\) , \(0 < \beta \le p -1\)和\(g{:}\,[0,\infty ) \rightarrow [0,\infty )\)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\(\beta < p-1\)和\(1/\lambda _1\)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。