AI 中文总结
该研究针对含向量p&q-拉普拉斯型扩散算子的各向异性椭圆狄利克雷问题，建立比较原理，分析不同反应指数下解的存在性、非存在性等结果，结合多种变分方法完成论证。
AI 中文摘要
我们研究一类广泛的各向异性狄利克雷问题，该问题由非齐次算子$\boldsymbol{\nabla}_A u=-\boldsymbol{\nabla}_{i=1}^N\boldsymbol{\nabla}_i\bigl(a_i(|\boldsymbol{\nabla}_i u|^{p_i})|\boldsymbol{\nabla}_i u|^{p_i-2}\boldsymbol{\nabla}_i u\bigr)$驱动，其中函数$a_i$满足$(p_i,q_i)$型的结构增长与单调性条件。该框架包含各向异性$\boldsymbol{\nabla}p$-拉普拉斯、正交各向异性p&q-拉普拉斯及结合多种方向增长模式的各向异性算子等示例。我们首先建立适配该非齐次性的比较原理；对于含奇异范围的幂非线性项，根据反应指数相对于各向异性增长指数与临界指数的位置，得到非存在性、存在性、奇异唯一性、比较替代关系及多重性结果。特别地，中间幂 regime 对小正参数表现为非存在性，对大参数则存在两个不同的非平凡解，且无需对最小有效指数附加额外限制；在部分共振$r=p_1<p_N$时，我们确定了一个精确的正参数阈值：参数不超过该阈值时无非平凡解，超过时则存在无穷多对强收敛于零的解。我们还处理次临界或临界增长的凹-凸反应，证明对应参数范围内存在非平凡非负弱解。论证结合了Díaz--Saa型凸性方法、各向异性Sobolev嵌入、直接极小化、Ekeland变分原理、山路技术及集中-紧性原理。
英文摘要
We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator \[ \mathcal L_Au=-\sum_{i=1}^N\partial_i\!\left( a_i(|\partial_i u|^{p_i})|\partial_i u|^{p_i-2} \partial_i u\right), \] where the functions $a_i$ satisfy structural growth and monotonicity conditions of $(p_i,q_i)$-type. This framework contains, among other examples, the anisotropic $\vec p$-Laplacian, the orthotropic $p\&q$-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, $r=p_1<p_N$, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave--convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz--Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration--compactness.
Comments32 pages
