Existence of three solutions for a two-point singular boundary-value problem with an unbounded weight

Abstract

We show the existence of three solution for the singular boundary-value problem $$\displaylines{ -z'' = h(t) \frac{f(z)}{z^\beta} \quad \text{in } (0,1) ,\cr z(t)> 0 \quad \text{in } (0,1),\cr z(0)= z(1)=0 , }$$ where $ 0 0$ and $0<\alpha<1-\beta$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $\psi_1\leq \psi_2\leq \phi_1, \psi_1\leq \phi_2\leq \phi_1 $ with $\psi_2 \not \leq \phi_2$, and $\psi_2 ,\phi_2$ are strict sub and super solutions. The establish the existence of at least three solutions $z_1, z_2, z_3$ satisfying $z_1\in [\psi_1,\phi_2]$, $z_2\in [\psi_2,\phi_1]$ and $z_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.

Keywords

• Three solutions theorem
• singular boundary value problems