Abstract
In this paper, we prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth: − Δ u = | u | 2 ⁎ − 2 u + g ( u ) in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a bounded domain in R N with C 3 boundary, N ⩾ 3 , ν is the outward unit normal of ∂ Ω, 2 ⁎ = 2 N N − 2 , and g ( t ) = μ | t | p − 2 t − t , or g ( t ) = μ t , where p ∈ ( 2 , 2 ⁎ ) , μ > 0 are constants. We obtain the existence of infinitely many solutions under certain assumptions on N, p and ∂ Ω. In particular, if g ( t ) = μ t with μ > 0 , N ⩾ 7 , and Ω is a strictly convex domain, then the problem has infinitely many solutions.
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