Nonhomogeneous quasilinear elliptic systems with small perturbations and lack of compactness

• Published in: Bulletin of mathematical sciences Vol. 15; no. 2
• Main Authors: Zhang, Xingyong, Qi, Wanting
• Format: Journal Article
• Language: English
• Published: World Scientific Publishing Company 01.08.2025; World Scientific Publishing
• Subjects: Clark’s theorem; locally symmetric conditions; Moser’s iteration; Quasilinear elliptic system; Clark’s theorem; Moser’s iteration; Quasilinear elliptic system; locally symmetric conditions
• ISSN: 1664-3607; 1664-3615
• DOI: 10.1142/S1664360725500043

We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by Stuart [Milan J. Math. 79 (2011) 327–341] and depends not only on ∇ u but also on u. We show 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 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 quasilinear elliptic system with nonhomogeneous differential operators and obtain 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.