Evolution of bifurcation curves for one-dimensional Minkowski-curvature problem

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This paper is concerned with the Robin problem for the prescribed mean curvature equation in Minkowski space −(u′∕1−|u′|2)′=λuq+μup,t∈(0,L),u′(0)=u(L)=0,(P) where 0<q<1<p. We show that there exists a constant μ∗>0 and two functions Λ∗(⋅),Λ∗(⋅) with Λ∗(μ)<0<Λ∗(μ),μ>μ∗, such that for every μ>μ∗ and all λ∈(Λ∗(μ),0), (P) has at least two positive solutions; for every μ>μ∗ and all λ∈(0,Λ∗(μ)), (P) has at least three positive solutions. The proof combines topological degree and bifurcation technique. We also present a numerical computation of the bifurcation curves.