On the Structure of the Solution Set of a Sign Changing Perturbation of the p-Laplacian under Dirichlet Boundary Condition

2013/10/21 by J.V. Gonçalves, J. V. Goncalves, Goncalves, J. V. +2
Computer Science · Mathematics · #35J25 #35J55 #35J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP #msc:35J25 #msc:35J55 #msc:35J70

paper · pdf · doi:10.48550/arxiv.1310.5636

23 pages

arxiv created 2013/10/21 · openalex publication_date 2013/10/21 · arxiv updated 2013/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a recent paper D. D. Hai showed that the equation -Δ_p u = λf(u) in Ω, under Dirichlet boundary condition, where Ω⊂ \bf R^N is a bounded domain with smooth boundary ∂Ω, Δ_p is the p-Laplacian, f : (0,∞) → \bf R is a continuous function which may blow up to ± ∞ at the origin, admits a solution if λ> λ₀ and has no solution if 0 < λ< λ₀. In this paper we show that the solution set S of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.

Citations

• Boundary regularity for solutions of degenerate elliptic equations
• On a singular nonlinear elliptic boundary-value problem
• C1 + α local regularity of weak solutions of degenerate elliptic equations

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