本人致力于运用非线性泛函分析的理论和方法研究分数阶或高阶椭圆型方程、Schrödinger方程与波方程,提出了“伸缩球面法”,并应用它对一般区域上各种不具平移不变性或具有奇性问题建立最优Liouville定理,克服了前人文献中建立Liouvill;提出了与非局部算子匹配的非局部球外平均及积分迭代估计方法,由此给出了任意阶共形不变方程和双非局部问题解的完整分类;建立了能量临界拟线性非局部问题解的最优渐近估计和径向对称性;得到了拟线性Hénon方程非径向解的存在性与渐近性质刻画;建立了完全非线性k-Hessian Lane-Emden方程的最优刚性结果。已主持国家自然科学基金青年科学基金B类等基金项目6项,负责国家科技重大专项子课题,已发表论文51篇(其中,ESI高被引3篇、热点论文1篇),发表在包括Proc. London Math. Soc.、Advances in Mathematics、Math. Annalen (2篇)、Trans. Amer. Math. Soc.、Analysis & PDE、Int. Math. Res. Not. (IMRN期刊Most Cited论文)、J. Funct. Anal.、SIAM J. Math. Anal. (2篇)、J. d'Analyse Math.、Calc. Var. & PDE (2篇)、Math. Z. (2篇)、Rev. Mat. Iberoam.、Science China: Math.等权威学术期刊。研究成果被C. Kenig (国际数学联盟前主席、ICM一小时报告人、美国科学院院士)、A. D. Ionescu (ICM45分钟报告人、美国科学院院士)、Y. Giga (ICM45分钟报告人、Math. Annalen主编)、J. Pipher (ICM45分钟报告人)e定理的三种常用方法的局限性、N. Masmoudi (ICM45分钟报告人、费马奖得主)、魏军城 (ICM45分钟报告人、加拿大皇家科学院院士)、辛周平 (ICM45分钟报告人、晨兴数学金奖得主)、杜一宏 (澳大利亚科学院院士)等著名学者引用、正面评价并应用于研究其它问题(他引超800次)。
Wei Dai; Lixiu Duan; Changfeng Gui; Yuan Li; Non-radial solutions for the critical quasi-linear H\'{e}non equation involving $p$-Laplacian in $\R^N$, Proc. London Math. Soc., 2026, 132 (4): e70148, 61 pp.
Wei Dai; Guolin Qin; Classification of nonnegative classical solutions to third-order equations, Advances in Mathematics, 2018, 328: 822-857.
Wei Dai; Yafei Li; Zhao Liu; Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to D^{1,p}-critical quasi-linear static Schrödinger-Hartree equation involving p-Laplacian $-\Delta_p$, Math. Ann., 2025, 391 (2): 2653-2708.
Daomin Cao; Wei Dai; Guolin Qin; Super poly-harmonic properties for nonnegative solutions to equations involving higher-order fractional Laplacians and its applications, Trans. Amer. Math. Soc., 2021, 374 (7): 4781-4813.
Wei Dai; Guolin Qin; Liouville type theorems for fractional and higher order H\'{e}non-Hardy type equations via the method of scaling spheres, Int. Math. Res. Not. IMRN, 2023, 2023 (11): 9001-9070.