Existence of nontrivial solutions for critical biharmonic equations with logarithmic term

2023/03/14 by He, Qihan, Lv, Juntao, Lv, Zongyan +1
#2020: 35A01 #35A15 #35B33 #35D30 #35G30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.07659

Abstract

In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begincases Δ² u=μΔu+λu+|u|^2^**-2u+τulog u², x∈Ω, u|_{∂ Ω}=(∂ u)/(∂ n)|_∂Ω=0, \endcases where μ,λ,τ∈ ℝ, |μ|+|τ|≠ 0, Δ²=ΔΔ denotes the iterated N-dimensional Laplacian, Ω⊂ ℝ^N is a bounded domain with smooth boundary ∂ Ω, 2^**=(2N)/(N-4)(N≥5) is the critical Sobolev exponent for the embedding H₀²(Ω)\hookrightarrow L^2^**(Ω) and H₀² (Ω) is the closure of C₀^ ∞ (Ω) under the norm || u ||:=(∫_Ω|Δu|²)^(1)/(2). The uncertainty of the sign of slog s² in (0,+∞) has some interest in itself. To know which of the three terms μΔu, λu and τu log u² has a greater influence on the existence of nontrivial weak solutions, we prove the existence of nontrivial weak solutions to the above problem for N≥5 under some assumptions of λ, μ and τ.

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