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

Abstract

In this article, we prove the existence of eigenvalues for the problem (ϕp(u′(t)))′+λh(t)ϕp(u(t))=0,t∈(0,1),Au(0)−A′u′(0)=0,Bu(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 ϕp(s)=∣s∣p−2s,p>1{\phi }_{p}\left(s)={| s| }^{p-2}s,p\gt 1, and hh 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: (ϕp(u′(t)))′+λh(t)f(u(t))=0,t∈(0,1),u(0)=0,au′(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≥0a\ge 0, f∈C((0,∞),(0,∞))f\in C\left(\left(0,\infty ),\left(0,\infty )) satisfies infs∈(0,∞)f(s)>0{\inf }_{s\in \left(0,\infty )}f\left(s)\gt 0 and limsups→0sαf(s)0\alpha \gt 0.

Keywords

• p-laplacian
• spectrum
• nonlinear boundary conditions
• bifurcation
• 34b18
• 34c10
• 34c23