Positive Solutions for Singular Superlinear $\phi$-Laplacian Problems with Nonlinear Boundary Conditions

Abstract

We prove the existence, nonexistence and multiplicity of positive solution to the problem

$$\begin{aligned} \left\{ \begin{array}{c} -(\phi (u^{\prime }))^{\prime }=\lambda h(t)f(u),\ 0<t<1, \\ u(0)=0,\ u^{\prime }(1)+H(u(1))=0, \end{array} \right. \end{aligned}$$

where $\phi$ is an odd, increasing homeomorphism on ${\mathbb {R}},\ h:(0,1]\rightarrow [0,\infty ),\ H:[0,\infty )\rightarrow [0,\infty )$ is nondecreasing,$\ f:(0,\infty )\rightarrow (0,\infty )$ is continuous, $\phi$-superlinear at $\infty$ with possible singularity at 0,  and $\lambda$ is a positive parameter.