Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions

2025/04/21 by Bhakta, Mousomi, Biswas, Nirjan, Das, Paramananda · 1 citation
#35B09 #35B33 #35J20 #35J25 #35J62 #35M12 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.15000

Abstract

We study the existence and multiplicity of positive solutions for the following concave-critical problem driven by an operator of mixed order obtained by the sum of the classical p-Laplacian and of the fractional p-Laplacian, P_λ,ε -Δ_p u+ε(-Δ_p)^s u=λ|u|^q-2u+|u|^{p^*-2}u in Ω, u=0 in ℝ^N ∖ Ω, where Ω⊂ℝ^N is a bounded open set, ε∈(0,1], 00, we prove Ambrosetti-Brezis-Cerami type results. In particular, we prove the existence of Λ_ε such that (P_λ,ε) has a positive minimal solution for 0<λ<Λ_ε, a positive solution for λ=Λ_ε and no positive solution for λ>Λ_ε. We also prove the existence of 0<λ^#≤Λ_ε such that (P_λ,ε) has at least two positive solutions for λ∈(0,λ^#) provided ε small enough. This extends the recent result of Biagi and Vecchi (Nonlinear Anal. 256 (2025),113795), Amundsen, et al. (Commun. Pure Appl. Anal., 22(10):3139-3164, 2023) from p=2 to the general 1

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• Mixed local-nonlocal equations with critical nonlinearity on ℝ^N: Non-existence, Existence, and Multiplicity of positive solutions

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