AI 中文总结
该研究针对一类混合各向异性与非局部奇异拟线性椭圆问题，采用单调逼近等方法证明弱解存在唯一性，刻画相关Sobolev型不等式最佳常数并证明其可达性，结果在p=2的混合加权拉普拉斯情形下为新成果。
AI 中文摘要
我们研究一类与Muckenhoupt权相关的混合各向异性与非局部奇异拟线性椭圆问题。奇异非线性项在原点附近爆破，且其与各向异性、加权退化、非局部扩散的相互作用带来了重大分析挑战。我们采用单调逼近、加权Sobolev嵌入、紧性及变分方法，在数据的适当假设下建立弱解的存在性与唯一性。此外，我们刻画了相关混合各向异性与非局部加权Sobolev型不等式的最佳常数，证明该常数可达，并表明归一化弱解是唯一极值元。这些结果在混合加权拉普拉斯情形\boldsymbol{p=2}时仍是新的。
英文摘要
We consider a class of mixed anisotropic and nonlocal singular quasilinear elliptic problem associated with Muckenhoupt weights. The presence of the singular nonlinearity, which blows up near the origin along with its interaction with anisotropy, weighted degeneracy, and nonlocal diffusion creates significant analytical challenges. We employ monotone approximation, weighted Sobolev embeddings, compactness, and variational methods to establish the existence and uniqueness of weak solutions under suitable assumptions on the datum. Further, we characterize the best constant in an associated weighted mixed anisotropic and nonlocal Sobolev-type inequality, prove that it is attained, and show that the normalized weak solution is the unique extremal. These results are new even in the mixed weighted Laplace case \(p=2\).
Comments27 pages, comments are welcome
