Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight

Abstract

Abstract In this article, we prove the existence of eigenvalues for the problem ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) ϕ p ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , A u ( 0 ) − A ′ u ′ ( 0 ) = 0 , B u ( 1 ) + B ′ u ′ ( 1 ) = 0 \left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t){\phi }_{p}\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ Au\left(0)-A^{\prime} u^{\prime} \left(0)=0,\hspace{1em}Bu\left(1)+B^{\prime} u^{\prime} \left(1)=0\end{array}\right. under hypotheses that ϕ p ( s ) = ∣ s ∣ p − 2 s , p > 1 {\phi }_{p}\left(s)={| s| }^{p-2}s,p\gt 1 , and h h is a nonnegative measurable function on ( 0 , 1 ) \left(0,1) , which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem: ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 , \left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t)f\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ u\left(0)=0,\hspace{1em}au^{\prime} \left(1)+c\left(\lambda ,u\left(1))=0,\end{array}\right. where λ \lambda is a real parameter, a ≥ 0 a\ge 0 , f ∈ C ( ( 0 , ∞ ) , ( 0 , ∞ ) ) f\in C\left(\left(0,\infty ),\left(0,\infty )) satisfies inf s ∈ ( 0 , ∞ ) f ( s ) > 0 {\inf }_{s\in \left(0,\infty )}f\left(s)\gt 0 and limsup s → 0 s α f ( s ) < ∞ {\mathrm{limsup}}_{s\to 0}{s}^{\alpha }f\left(s)\lt \infty for some α > 0 \alpha \gt 0 .