Positive solutions of three-point boundary value problems for higher-order \(p\)-Laplacian with infinitely many singularities
Summary: We study the higher-order quasilinear equation with \(p\)-Laplacian \[ (\phi_p(u^{(n-1)}))'+g(t)f(t,u,u',\dots,u^{(n-2)})=0,\quad 0<t<1,\;n\geq3, \] subject to the following three-point boundary conditions: \[ \begin{aligned} u^i(0) = 0,\quad & 0\leq i\leq n - 3,\\ \alpha\phi_p(u^{(n-2)}(0))&-\beta\phi_p(u^{(n-1)}(\eta))=0,\\ \gamma\phi_p(u^{(n-2)}(1))&-\delta\phi_p(u^{(n-1)}(1))=0,\end{aligned} \] where \(\phi_p(s)\) is a \(p\)-Laplacian operator, that is, \(\phi_p(s)= |s|^{p-2}s\), \(p > 1\), \(\eta\in (0,1)\) is a given constant, \(\alpha > 0\), \(\gamma > 0\), \(\beta\geq 0\), \(\delta\geq 0\), \(g : [0,1] \to [0,\infty)\) has countable many singularities on \((0,1/2)\). We show that there exist countable many positive solutions by using the fixed point index theorem for operators in a cone.
scientific article; zbMATH DE number 5304374