On the existence of positive solution for a Neumann problem with double critical exponents in half-space

2024/04/06 by Yinbin Deng, Deng, Yinbin, Longge Shi +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2404.04488

openalex publication_date 2024/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the existence and nonexistence of positive solution for a Neumann problem with double critical exponents and fast increasing weighted in half-space. This problem is closely related to the study of self-similar solutions for nonlinear heat equation. By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for the Mountain Pass level, we obtain the existence of a positive solution under different assumptions. Meanwhile, some nonexistence results for this problem is also obtained by an improved Pohozaev identity and Hardy inequality according to the value of the parameters. Particularly, we give the best lower bound of the parameter for the existence of a positive solution of this problem if dimension N=4.

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