Threshold phenomena for least energy solutions of a doubly critical Neumann problem with a critical absorption term
Abstract
We investigate the existence and nonexistence of least energy solutions for a doubly critical Neumann problem with a critical absorption term. We consider a positive function u defined on a smooth bounded domain Ω⊂ℝn with n≥ 5, satisfying −Δu + λu = u2* − 1 −αu2♯−1 inside Ω, with Neumann boundary condition ∇ u· ν= u2♯ −1 on ∂Ω, where λ>0, α≥0, 2*=2n/(n−2) is the critical Sobolev exponent and 2♯=(2(n−1))/(n−2) denotes the critical trace exponent. The simultaneous presence of the interior and boundary critical exponents together with this critical absorption term creates a new interaction between competing concentration mechanisms. In particular, the absorption term has the same critical trace exponent as the boundary nonlinearity, but with the opposite sign and acting in the interior of Ω. Consequently, it competes with the boundary mean curvature correction at the same asymptotic order in the energy expansion, leading to a sharp threshold phenomenon. Combining several analytic techniques and geometric tools, we prove the existence of a threshold value α0=α0(λ,Ω)∈(0,+∞) such that the problem admits a least energy solution if α<α0, and no least energy solution if α>α0. Moreover, the problem also has a least energy solution at α=α0 provided α0> C(n)max∂ Ω H, where C(n) is a positive constant depending only on n and H is the mean curvature on ∂Ω.
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BibTeX
@misc{deng2026threshold,
title = {{Threshold phenomena for least energy solutions of a doubly critical Neumann problem with a critical absorption term}},
author = {Yinbin Deng and Yulin Shi and Pengyan Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07411},
eprint = {2609.07411},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07411},
}
RIS
TY - UNPB
TI - Threshold phenomena for least energy solutions of a doubly critical Neumann problem with a critical absorption term
AU - Yinbin Deng
AU - Yulin Shi
AU - Pengyan Wang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07411
UR - https://arxiv.org/abs/2609.07411
AB - We investigate the existence and nonexistence of least energy solutions for a doubly critical Neumann problem with a critical absorption term. We consider a positive function u defined on a smooth bounded domain Ω⊂ℝ^n with n≥ 5, satisfying −Δu + λu = u^2^* − 1 −αu^2^♯−1 inside Ω, with Neumann boundary condition ∇ u· ν= u^2^♯ −1 on ∂Ω, where λ>0, α≥0, 2^*=2n/(n−2) is the critical Sobolev exponent and 2^♯=(2(n−1))/(n−2) denotes the critical trace exponent. The simultaneous presence of the interior and boundary critical exponents together with this critical absorption term creates a new interaction between competing concentration mechanisms. In particular, the absorption term has the same critical trace exponent as the boundary nonlinearity, but with the opposite sign and acting in the interior of Ω. Consequently, it competes with the boundary mean curvature correction at the same asymptotic order in the energy expansion, leading to a sharp threshold phenomenon. Combining several analytic techniques and geometric tools, we prove the existence of a threshold value α_0=α_0(λ,Ω)∈(0,+∞) such that the problem admits a least energy solution if α<α_0, and no least energy solution if α>α_0. Moreover, the problem also has a least energy solution at α=α_0 provided α_0> C(n)max_∂ Ω H, where C(n) is a positive constant depending only on n and H is the mean curvature on ∂Ω.
ER -