Abstract
In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation \t\t\t{−(u′1−u′2)′=λf(u),x∈(−L,L),u(−L)=0=u(L),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} - (\\frac{u'}{\\sqrt{1-u^{\\prime \\,2}}} )'=\\lambda f(u), &x\\in (-L,L), \\\\ u(-L)=0=u(L), \\end{cases} $$\\end{document} where λ and L are positive parameters, fin C[0,infty ) cap C^{2}(0,infty ), and f(u)>0 for 0< u< L. We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f''(u)>0 and uf'(u)geq f(u)+frac{1}{2}u^{2}f''(u) for 0< u< L. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.
Highlights
1 Introduction In this work, we study the bifurcation curves and exact multiplicity of positive solutions of the quasilinear two-point boundary value problem
If f (|x|, u) = μ(|x|)up, p > 1 and μ : [0, ∞) → R is continuous, strictly positive on (0, ∞), Bereanu et al [2] obtained that there exists > 0 such that problem (1.2) has zero, at least one, or at least two positive radial solutions according to λ ∈ (0, ), λ =, or λ >
There exists λ∗ > 0 such that problem (1.1) has zero, exactly one, or exactly two positive solutions according to λ ∈ (0, λ∗), λ = λ∗, or λ > λ∗
Summary
We study the bifurcation curves and exact multiplicity of positive solutions of the quasilinear two-point boundary value problem. Coelho et al [4] proved the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. If f (|x|, u) = μ(|x|)up, p > 1 and μ : [0, ∞) → R is continuous, strictly positive on (0, ∞), Bereanu et al [2] obtained that there exists > 0 such that problem (1.2) has zero, at least one, or at least two positive radial solutions according to λ ∈ (0, ), λ = , or λ >. In this paper, we study the bifurcation curves and exact multiplicity of positive solutions of problem (1.1), where the nonlinearity f satisfies (H1) f ∈ C([0, ∞), R) and f (u) > 0 for 0 < u < L;.
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