Existence of positive solutions for multi-point boundary value problem with strong singularity
This paper is devoted to the study of the Sturm-Liouville boundary value problem \[ (\varphi_p(u'(t)))'+h(t)f(t,u(t))=0\quad \text{a.e. }\,0<t<1 \] with boundary conditions \[ u(0)=\sum_{i=1}^{m-2}a_iu(\xi_i),\;u(1)=\sum_{i=1}^{m-2}b_iu(\xi_i), \] where \(\varphi_p(s)=|s|^{p-2}s\,(p>1)\) is the \(p\)-Laplacian, \(0<\xi_1<\xi_2<\dots<\xi_{m-2}\) are points and \(\sum_{i=1}^{m-2}a_i<1,\;\sum_{i=1}^{m-2}b_i<1\). \(h\) is a nonnegative measurable function which may be singular at \(t=0\) and/or \(t=1\). The nonlinearity \(f\in C([0,1]\times[0,\infty),[0,\infty))\) does not necessarily satisfy the Carathéodory but some assumptions on the ratio \(\frac{f(t,u)}{\varphi_p(u)}\) are assumed. Existence of single and multiple solutions is then obtained via computation of the fixed point index on cones of Banach spaces. The paper ends with an example.
scientific article; zbMATH DE number 5800315
scientific article