Positive solutions of critical Hardy-Hénon equations with logarithmic term

2025/04/28 by He, Qihan, Liu, Wenxuan, Pan, Yiqing
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.19817

Abstract

We consider the existence, non-existence and multiplicity of positive solutions to the following critical Hardy-Hénon equation with logarithmic term \ -Δu =|x|^α|u|^{2^*_α-2}⋅ u+μulog u²+λu, · amp;x∈ Ω,
u=0, · amp;x∈ ∂ Ω, . where Ω=B for α≥ 0, Ω=B∖\0\ for α∈(-2,0), B⊂ℝ^N is an unit ball, λ, μ∈ ℝ, N≥ 3, α>-2, 2^*_α:=(2(N+α))/(N-2) is the critical exponent for the embedding H_0,r¹( Ω)\hookrightarrow L^p( Ω;|x|^α), and which can be seen as a Brézis-Nirenberg problem. When N ≥ 4 and μ>0, we will show that the above problem has a positive Mountain pass solution, which is also a ground state solution. At the same time, when μ<0, under some assumptions on the N, μ, λ and α, we will show that the above problem has at least a positive least energy solution and at least a positive Mountain pass solution, respectively. What's more, when certain inequality related to N ≥ 3, μ<0 and α∈(-2,0] holds, we will demonstrate the non-existence of positive solutions to the above-mentioned problem. The presence of logarithmic term brings some new and interesting phenomena to this problem.

Related

• Existence and multiplicity of positive solutions to a critical elliptic equation with logarithmic perturbation
• The existence of ground state solutions for critical Hénon equations in ℝ^N
• The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
• Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in ℝⁿ
• Mountain pass solution to the Brézis-Nirenberg problem with logarithmic perturbation