2025/01/03 by Zhang, Xingyong, Qi, Wanting
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.01602
We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only ∇ u but also u. We show that the existence of multiple small solutions when the nonlinear term F(x,u,v) satisfies locally sublinear and symmetric conditions and the perturbation is any continuous function with a small coefficient and no any growth hypothesis. Our technical approach is mainly based on a variant of Clark's theorem without the global symmetric condition. We develop the Moser's iteration technique to this quasi-linear elliptic system with nonhomogeneous differential operators and obtain that the relationship between ‖u‖∞, ‖v‖∞ and ‖u‖2∗, ‖v‖2∗. We overcome some difficulties which are caused by the nonhomogeneity of the differential operator and the lack of compactness of the Sobolev embedding.