2024/01/28 by Yinbin Deng, Deng, Yinbin, Longge Shi +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2401.15637
openalex publication_date 2024/01/28 · openalex created_date 2024/01/31 · openalex updated_date 2026/07/28
In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \ \beginaligned -Δu-(1)/(2)(x ⋅∇ u)amp;= λ|u|^2*-2u+μ|u|p-2uamp; in ℝN+, (∂ u)/(∂ n)amp;=√λ|u|^2*-2u amp; on ∂ ℝN+, \endaligned . where ℝN+=\(x', xN): x'∈ ℝN-1, xN>0\, N≥3, λ>0, μ∈ ℝ, 2< p <2*, n is the outward normal vector at the boundary ∂ ℝN+, 2*=(2N)/(N-2) is the usual critical exponent for the Sobolev embedding D1,2(ℝN+)\hookrightarrow L^2*(ℝN+) and 2*=(2(N-1))/(N-2) is the critical exponent for the Sobolev trace embedding D1,2(ℝN+)\hookrightarrow L^2*(∂ ℝN+). By establishing an improved Pohozaev identity, we show that the problem has no nontrivial solution if μ≤ 0; By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution for all λ>0 and the different values of the parameters p and μ>0. Particularly, for λ>0, N≥ 4, 20. Moreover, the existence of multiple solutions for the problem is also obtained by dual variational principle for all μ>0 and suitable λ.