On the existence of positive solution for a Neumann problem with double critical exponents in half-space

• Published in: Journal of Differential Equations 2024-08, Vol.401, p.376-427
• Main Authors: Deng, Yinbin, Shi, Longge
• Format: Article
• Language: English
• Subjects: Critical exponents; Half-space; Neumann problem; Self-similar solutions
• ISSN: 0022-0396
• DOI: 10.1016/j.jde.2024.05.002

In this paper, we consider the existence and nonexistence of positive solution for the following critical Neumann problem(0.1){−Δu−12(x⋅∇u)=λu+a|u|2⁎−2uinR+N,∂u∂n=μ|u|q−2u+|u|2⁎−2uon∂R+N, where R+N={(x′,xN):x′∈RN−1,xN>0} is the upper half-space, N≥3, λ,μ∈R are parameters, a∈{0,1}, n is the outward normal vector at the boundary ∂R+N, 2≤q0 and q∈(2,2⁎), we obtain that problem (0.1) has a positive solution if and only if λ∈[0,N2); On the other hand, for N≥3 and μ=0, we find a lower bound Λ⁎∈[N4,N2) depending on N such that problem (0.1) has a positive solution if λ∈(Λ⁎,N2) and has no positive solution if λ∈(−∞,N4)∪[N2,+∞). Moreover, we estimate that Λ⁎∈(N4,N−22) if N≥5 and Λ⁎=N4=1 if N=4 which gives the best lower bound of λ for the existence of a positive solution of problem (0.1) if N=4.