Infinitely many solutions for quasilinear systems with critical exponent. (October 2018) - Guo, Yuxia; Nie, Jianjun
Abstract: In this paper, we consider the following quasilinear system: (P) − ∑ i, j = 1 N D j ( a i j ( u ) D i u ) + 1 2 ∑ i, j = 1 N D s a i j ( u ) D i u D j u = a u + | u | 2 ∗ − 2 u + α 2 ∗ | u | α − 2 u | v | β in Ω, − ∑ i, j = 1 N D j ( a i j ( v ) D i v ) + 1 2 ∑ i, j = 1 N D s a i j ( v ) D i v D j v = b v + | v | 2 ∗ − 2 v + β 2 ∗ | u | α | v | β − 2 v in Ω, u = v = 0 on ∂ Ω, where D i = ∂ ∂ x i, D s a i j ( s ) = d d s a i j ( s ), a, b > 0 are constants, α, β > 1 and α + β = 2 ∗, 2 ∗ = 2 N N − 2 is the Sobolev critical exponent for the embedding of H 0 1 ( Ω ) into L p ( Ω ), Ω ⊂ R N is an open bounded domain with smooth boundary. By accurate estimates of the concentration compactness and the asymptotic behavior of the approximation solutions to the subcritical problems, we prove that if N ≥ 7, the problem ( P ) admits infinitely many solutions.