p(x)-Laplacian-Like Neumann Problems in Variable-Exponent Sobolev Spaces Via Topological Degree Methods

2021/12/12 by Mohamed El Ouaarabi, Ouaarabi, Mohamed El, Chakir Allalou +3
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.2112.06262

openalex publication_date 2021/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the existence of a "weak solutions" for a Neumann problems of p(x)-Laplacian-like operators, originated from a capillary phenomena, of the following form \-\rmdiv(\vert∇ u\vert^p(x)-2∇ u+\frac\vert∇ u\vert^2p(x)-2∇ u√1+\vert∇ u\vert^2p(x))=λf(x, u, ∇ u) · amp; in Ω,
(\vert∇ u\vert^p(x)-2∇ u+\frac\vert∇ u\vert^2p(x)-2∇ u√1+\vert∇ u\vert^2p(x))(∂ u)/(∂η)=0 · amp; on ∂Ω,. in the setting of the variable-exponent Sobolev spaces W^1,p(x)(Ω), where Ω is a smooth bounded domain in ℝ^N, p(x)∈ C₊(Ω) and λ is a real parameter. Based on the topological degree for a class of demicontinuous operators of generalized (S₊) type and the theory of variable-exponent Sobolev spaces, we obtain a result on the existence of weak solutions to the considered problem.

Related

• Existence and Uniqueness of Weak solution of p( x ) - laplacian in\n Sobolev spaces with variable exponents in complete manifolds
• Solutions for Neumann boundary value problems involving (p₁(x), p₂(x))-Laplace operators
• On a nonlinear eigenvalue problem in Sobolev spaces with variable exponent
• Nonlinear problem involving the fractional p(x)-Laplacian operator by Topological degree
• Nonlinear eigenvalue problems in Sobolev spaces with variable exponent