Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

Abstract

Abstract In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ where λ and L are positive parameters, f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ , and f ( u ) > 0 $f(u)>0$ for 0 0 $f''(u)>0$ and u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ for 0 < u < L $0< u< L$ . 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.

Keywords

• Minkowski-curvature equation
• Exact multiplicity
• Positive solution
• Bifurcation curves
• Time map