奇异正则化$p(x)$-Laplacian Dirichlet问题的正归一化解
Positive normalized solutions for a singular regularized $p(x)$-Laplacian Dirichlet problem
浏览论文内容
中文总结 AI 辅助
该文研究带模约束的奇异正则化p(x)-Laplacian Dirichlet问题，通过构造非奇异逼近泛函的约束极小元，利用变分方法和Hardy不等式，证明了正归一化弱解的存在性。
中文摘要 AI 辅助
我们研究了一个由正则化$p(x)$-Laplacian驱动的奇异Dirichlet问题，并施加了规定的模约束。利用正则化能量的增长性和可微性估计，结合强制性和模紧性，我们为一族非奇异逼近泛函构造了非负约束极小元。在允许有符号约束变分的条件下，每个极小元满足一个带有唯一确定的约束乘子的近似Euler-Lagrange方程。一个显式的均匀边界下障碍函数和一个相容的权函数条件，结合变指数Hardy不等式，提供了对奇异源的全局对偶控制，并确保对所有零迹Sobolev检验函数的收敛性。对于这些近似极小对，奇异极限结果给出了状态的子序列强Sobolev收敛性以及乘子的收敛性，同时保持了正性和规定的模。因此，极限对是奇异问题的正归一化弱解，该解通过移除反应正则化而保持梯度正则化不变获得。
英文摘要
We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating functionals. Under the admissibility of signed constrained variations, each minimizer satisfies an approximate Euler--Lagrange equation with a uniquely determined constraint multiplier. An explicit uniform boundary lower barrier and a compatible weight condition, combined with a variable exponent Hardy inequality, provide global dual control of the singular sources and convergence against every zero-trace Sobolev test function. For these approximate minimizing pairs, the singular-limit result gives subsequential strong Sobolev convergence of the states and convergence of their multipliers, while preserving positivity and the prescribed modular. The limiting pair is therefore a positive normalized weak solution of the singular problem, obtained by removing the reaction regularization while keeping the gradient regularization fixed.
