各向异性拟线性椭圆方程的非负分布解

Nonnegative Distributional Solutions to Anisotropic Quasilinear Elliptic Equations

摘要: 本文聚焦一类具有退化主部与低可积性$L^1$ 源项的非强制各向异性拟线性椭圆方程，探究其解的存在性及特定Sobolev空间的正则性。研究区域为$R^N\left(N\ge 3\right)$ 中的有界$C^{1,1}$ 区域，方程主部含退化因子，衰减率$0<r<\overline{p}-1$ ，且非线性项在零点附近和无穷远处表现为不同的奇异行为。通过构建“近似正则化–先验估计–弱收敛过渡”分析框架，利用截断测试函数、Hölder不等式及Sobolev嵌入定理，得到方程在特定Sobolev空间存在非负分布解。

Abstract: This paper focuses on a class of non-coercive anisotropic quasilinear elliptic equations with degenerate principal parts and low-integrability $L^1$ source terms, investigating the existence of their solutions and the regularity in specific Sobolev spaces. The research domain is a bounded $C^{1,1}$ domain in $R^N\left(N\ge 3\right)$ . The principal part of the equation contains a degenerate factor with a decay rate satisfying $0<r<\overline{p}-1$ , and the nonlinear terms exhibit distinct singular behaviors near zero and at infinity. By constructing an analytical framework of “approximate regularization-a priori estimates-weak convergence transition” and utilizing truncated test functions, Hölder’s inequality, and Sobolev embedding theorems, it is shown that there exist non-trivial distributional solutions to the equation in specific Sobolev spaces.