Singular problems: An upper and lower solution approach
Consider the problem \[ (py')'+p(t)q(t)f(t,y)=0, \quad \lim_{t\to 0+}p(t)y'(t)=0,\;y(1)=0, \tag{1} \] where \(p\) can be zero at both end points \(0\) and \(1\), \(q\) can be singular at these points and \(f\in{\mathcal C}([0,1]\times (0,\infty))\) can have a singularity at \(y=0\). The authors assume the existence of a sequence of constants \(\rho_n\) which tend to zero as \(n\) goes to infinity and are lower functions for (1) on \([\frac{1}{n},1]\), a function \(\alpha\), which is a strict lower solution to (1), where the nonlinearity is modifed for \(t\in [0, \frac{1}{n}]\), a sequence of upper solutions \(\beta_n\geq\max\{\alpha,\rho_n\}\) of the same modified problems. The main result of the paper states conditions to ensure, within such a framework, the existence of solutions to (1). This result applies to the problem \[ (t^3y')'+t^2(\frac{1}{\sqrt{y}}-\mu)=0, \quad \lim_{t\to 0+}t^3y'(t)=0,\;y(1)=0, \] with \(\mu>0\). The idea of the proof is to approximate the problem by a sequence of nonsingular problems. For each of them Schauder's theorem applies and the result follows then from Arzelà-Ascoli's theorem. Extensions are worked out for the derivative dependent case \[ (py')'+p(t)q(t)f(t,y,py')=0, \quad \lim_{t\to 0+}p(t)y'(t)=0,\;y(1)=0, \] as well as for the Dirichlet problem \[ (py')'+p(t)q(t)f(t,y,py')=0, \quad y(0)=0,\;y(1)=0. \] In this last case, the authors assume \(1/p\in L^1(0,1)\) so that the Liouville transformation applies.
scientific article; zbMATH DE number 1545044