Existence and multiplicity of solutions to perturbed Δα1,β1α,β-Laplacian system in RN involving critical nonlinearity

• Published in: Rendiconti del Circolo matematico di Palermo 2024, Vol.73 (7), p.2749-2765
• Main Authors: Hanh, Le Thi Hong, Luyen, Duong Trong, Thuy, Pham Thi
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Nonlinearity; Standing waves
• ISSN: 0009-725X; 1973-4409
• DOI: 10.1007/s12215-024-01070-y

In this article, we consider a Δ α 1 , β 1 α , β -Laplacian system with critical nonlinearity in R N - ε 2 Δ α 1 , β 1 α , β u + b ( X ) u = g ( X ) | u | 2 ~ ∗ - 2 u + F u ( X , u , v ) , X ∈ R N , - ε 2 Δ α 1 , β 1 α , β v + b ( X ) v = g ( X ) | v | 2 ~ ∗ - 2 v + F v ( X , u , v ) , X ∈ R N , u ( X ) , v ( X ) → 0 as | X | → ∞ , where Δ α 1 , β 1 α , β is the subelliptic operator of the type Δ α 1 , β 1 α , β : = Δ x + Δ y + x 2 α y 2 β x α 1 + y β 1 2 Δ z ; x ∈ R N 1 ; y ∈ R N 2 , z ∈ R N 3 ; N = N 1 + N 2 + N 3 , α , β , α 1 , β 1 ≥ 0 , N ~ : = N 1 + N 2 + N 3 ( 1 + α + α 1 + β + β 2 ) , 2 ~ ∗ = 2 N ~ / ( N ~ - 2 ) , X = ( x , y , z ) , ( N ~ > 2 ) . Under some proper conditions, we obtain the existence of standing wave solutions ( u ε , v ε ) which tend to the trivial solutions as ε → 0 . Moreover, we get m pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results.