2025/04/15 by Qiang Zhang, Zhang, Q., Yuzhu Han +1
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2504.10970
In this paper we give a positive answer to the conjecture raised by Hajaiej et al. (J. Geom. Anal., 2024, 34(6): No. 182, 44 pp) on the existence of a mountain pass solution at positive energy level to the Brézis-Nirenberg problem with logarithmic perturbation. To be a little more precise, by taking full advantage of the local minimum solution and some very delicate estimates on the logarithmic term and the critical term, we prove that the following problem \begincases -Δu= λu+μ|u|2u+θulog u2, amp;x∈Ω,
u=0, amp;x∈∂Ω\endcases possesses a positive mountain pass solution at positive energy level, where Ω⊂ ℝ4 is a bounded domain with smooth boundary ∂Ω, λ∈ ℝ, μ>0 and θ<0. A key step in the proof is to control the mountain pass level around the local minimum solution from above by a proper constant to ensure the local compactness. Moreover, this result is also extended to three-dimensional and five-dimensional cases.