A non-trivial solution for a p-Schrödinger–Kirchhoff-type integro-differential system by non-smooth techniques

• Published in: Annals of functional analysis 2023-10, Vol.14 (4), Article 77
• Main Authors: Mayorga-Zambrano, Juan, Narváez-Vaca, Daniel
• Format: Article
• Language: English
• Subjects: Functional Analysis; Mathematics; Mathematics and Statistics; Original Paper
• ISSN: 2639-7390; 2008-8752
• DOI: 10.1007/s43034-023-00299-5

We consider the integro-differential system ( P m ) : - a k + b k ∫ R N | ∇ u k | p d x p - 1 Δ p u k + V ( x ) | u k | p - 2 u k = ∂ k F ( u 1 , … , u m ) , where x ∈ R N , a k > 0 , b k ≥ 0 , N ≥ 2 and 1 < p < N , u k ∈ W 1 , p ( R N ) , for k = 1 , … , m . By ∂ k F ( u 1 , … , u m ) , it is denoted the k -th partial generalized gradient in the sense of Clarke. The potential V ∈ C R N verifies inf ( V ) > 0 and a coercivity property introduced by Bartsch et al. The coupling function F : R m ⟶ R is locally Lipschitz and verifies conditions introduced by Duan and Huang. By applying tools from the non-smooth critical point theory, we prove the existence of a non-trivial mountain pass solution of ( P m ) .