Normalized solution for p-Laplacian equation in exterior domain

2024/07/16 by Wei-Qiang Zhang, Zhang, Weiqiang, Yanyun Wen +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.11415

openalex publication_date 2024/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are devoted to the study of the following nonlinear p-Laplacian Schrödinger equation with L^p-norm constraint \begincases amp;-Δ_p u=λ|u|^p-2u +|u|^r-2u\quadin Ω,
amp;u=0\quadon ∂Ω,
amp;∫_Ω|u|^pdx=a, \endcases where Δ_pu=div (|∇ u|^p-2∇ u), Ω⊂ℝ^N is an exterior domain with smooth boundary ∂Ω≠∅ satisfying that \R^N∖Ω is bounded, N≥3, 2≤ p0 and λ∈\R is an unknown Lagrange multiplier. First, by using the splitting techniques and the Gagliardo-Nirenberg inequality, the compactness of Palais-Smale sequence of the above problem at higher energy level is established. Then, exploiting barycentric function methods, Brouwer degree and minimax principle, we obtain a solution (u,\la) with u>0 in \R^N and \la<0 when \R^N∖Ω is contained in a small ball. Moreover, we give a similar result if we remove the restriction on Ω and assume a>0 small enough. Last, with the symmetric assumption on Ω, we use genus theory to consider infinite many solutions.

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