The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem ${\left(\phi \left(u^{\text{'}}\right)\right)}^{\text{'}}=\lambda f(t,u,u^{\text{'}})$, $u\left(0\right)=u\left(T\right)=A$. Here $\lambda$ is the positive parameter, $A>0$, $f$ is singular at the value $0$ of its first phase variable and may be singular at the value $A$ of its first and at the value $0$ of its second phase variable.

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems$$\left\{\begin{array}{cc}-{\left[\phi \left(u^{\text{'}}\right)\right]}^{\text{'}}=\lambda u^p\left(1-{\displaystyle \frac{u}{N}}\right)\hfill & \text{in}(-L,L),\hfill \\ u(-L)=u\left(L\right)=0,\hfill \end{array}\right.$$ where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi \left(u\right)$ is either $\phi \left(u\right)=u$ or $\phi \left(u\right)=u/\sqrt{1-u^2}$. We prove that the corresponding bifurcation curve is $\subset$-shape. Thus, the exact multiplicity of positive solutions can be obtained.

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ${\left({\phi }_p\left(u^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\right)=0$, $0<t<1$, where ${\phi }_p\left(s\right):={\left|s\right|}^{p-2}s$, $p>1$, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient $q\left(t\right)$ may be singular at $t=0,1$.

We prove the existence of a positive solution to the BVP $${\left(\Phi \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=f(t,u\left(t\right)),u^{\text{'}}\left(0\right)=u\left(1\right)=0,$$ imposing some conditions on Φ and f. In particular, we assume $\Phi \left(t\right)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_{\infty }$ bound for the solution is provided by the $L_{\infty }$ norm of any test function with negative energy.