Existence and multiplicity of positive solutions for singular $\phi$-Laplacian superlinear problems with nonlinear boundary conditions

Abstract

We prove the existence and multiplicity of positive solutions to the singular $\phi$-Laplacian BVP \begin{align*} \begin{cases} -(r(t)\phi(u′))′=\lambda g(t)(f(u)-(a/(u^{\alpha}))),& t\in(0,1),\\ u(0)=0, u′(1)+H(u(1))=0 \end{cases} \end{align*} for a certain range of the parameter $\lambda>0$, where $a>0$, $\alpha\in(0,1)$, $\phi$ is an odd, increasing and convex homeomorphism on $\mathbb{R}$, and $f$ is $\phi$-superlinear at $\infty$.

Keywords

• $\phi$-laplacian
• infinite semipositone
• positive solutions