Positive solutions for concave-convex type problems for the one-dimensional ϕ-Laplacian

2023/02/23 by Uriel Kaufmann, Kaufmann, Uriel, Leandro Milne +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #34B15 #34B18 #Bone Metabolism and Diseases #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #RNA Research and Splicing

paper · pdf · doi:10.48550/arxiv.2302.12350

openalex publication_date 2023/02/23 · openalex created_date 2023/03/01 · openalex updated_date 2026/07/28

Abstract

Let Ω=(a,b)⊂ℝ, 0≤ m,n∈ L¹(Ω), λ,μ>0 be real parameters, and ϕ:ℝ→ℝ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \begincases -ϕ( u^′) ^′=λm(x)f(u)+μn(x)g(u) amp; in Ω,
u=0 amp; on ∂Ω, \endcases where f,g:[0,∞)→\lbrack0,∞) are continuous functions which are, roughly speaking, sublinear and superlinear with respect to ϕ, respectively. Our assumptions on ϕ, m and n are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\uı fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.

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