A different approach to ground state solutions for p-Laplacian system with critical exponent. (January 2021) - Zhen, Maoding; Zhang, Binlin
Abstract: In this paper, we give a different method from Ao and Zou (2019) and Chen and Zou (2012) to consider the following nonlinear Schrödinger system with one critical exponent and one subcritical exponent: − Δ p u + μ | u | p − 2 u = | u | q − 2 u + α λ | u | α − 2 u | v | β in R N, − Δ p v + ν | v | p − 2 v = | v | p ∗ − 2 v + β λ | u | α | v | β − 2 v in R N, where N ≥ max { 3, p }, μ, ν, λ > 0, α ≥ 1, β ≥ 1, 2 ≤ p < q < p ∗ and α + β = p, p ∗ = N p ∕ ( N − p ) . By using variational methods, we prove that there exists μ 0 ∈ ( 0, 1 ), such that when 0 < μ ≤ μ 0, the above system has a positive ground state solution; when μ > μ 0, there exists λ μ, ν ∈ μ − μ 0 α α p ν β β p, ( μ α ) α p ( ν β ) β p such that if λ > λ μ, ν, the above system has a positive ground state solution, if λ < λ μ, ν, the above system has no ground state solution.