A unified approach to singular problems arising in the membrane theory

summary:We consider the singular boundary value problem $$ (t^nu'(t))'+ t^nf(t,u(t))=0, \quad \lim _{t\to 0+}t^nu'(t)=0, \quad a_0u(1)+a_1u'(1-)=A, $$ where $f(t,x)$ is a given continuous function defined on the set $(0,1]\times (0,\infty )$ which can have a time singularity at $t=0$ and a space singularity at $x=0$. Moreover, $n\in \Bbb N$, $n\ge 2$, and $a_0$, $a_1$, $A$ are real constants such that $a_0\in (0,\infty )$, whereas $a_1,A\in [0,\infty )$. The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation