Bifurcation curve for the Minkowski-curvature equation with concave or geometrically concave nonlinearity

Abstract

Abstract We study the bifurcation curve and exact multiplicity of positive solutions in the space C 2 ( ( − L , L ) ) ∩ C ( [ − L , L ] ) $C^{2}\left ( (-L,L)\right ) \cap C\left ( [-L,L]\right ) $ for the Minkowski-curvature equation { − ( u ′ ( x ) 1 − ( u ′ ( x ) ) 2 ) ′ = λ f ( u ) , − L 0 $\lambda >0$ is a bifurcation parameter, f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) $f\in C[0,\infty )\cap C^{2}(0,\infty )$ satisfies f ( u ) > 0 $f(u)>0$ for u > 0 $u>0$ and f is either concave or geometrically concave on ( 0 , ∞ ) $(0,\infty )$ . If f is a concave function, we prove that the bifurcation curve is monotone increasing on the ( λ , ∥ u ∥ ∞ ) $(\lambda ,\left \Vert u\right \Vert _{\infty })$ -plane. If f is a geometrically concave function, we prove that the bifurcation curve is either ⊂-shaped or monotone increasing on the ( λ , ∥ u ∥ ∞ ) $(\lambda ,\left \Vert u\right \Vert _{\infty })$ -plane under a mild condition. Some interesting applications are given.

Keywords

• Exact multiplicity
• Positive solution
• Bifurcation curve
• Minkowski-curvature equation
• Concave nonlinearity
• Geometrically concave nonlinearity