On the nonhomogeneous fourth-order \(p\)-Laplacian generalized Sturm-Liouville nonlocal boundary value problems
The authors consider the boundary value problem \[ \begin{aligned} (\phi_p(u''(t)))''- k^2\phi_p(u''(t))&= g(t) f(t,u(t)),\quad t\in (0,1),\\ u''(0)= \phi_q(a),&\quad u''(1)= \phi_q(b),\\ \alpha u(0)-\beta u'(0)&= \sum^{n-2}_{i=1} a_i u(\xi_i),\\ \gamma u(1)+\delta u'(1)&= \sum^{n-2}_{i=1} b_i u(\xi_i),\end{aligned}\tag{\(*\)} \] where \(\phi_p\) is the \(p\)-Laplacian operator, that is, \(\phi_p(u)=|u|^{p-2}u\), \(p> 1\), \(\phi^{-1}_p= \phi_q\), \(1/p+1/q= 1\), \(k\neq 0\), \(\alpha,\beta,\gamma,\delta\geq 0\), \(\xi_i\in (0,1)\), \(a,b,a_i,b_i\in (0,\infty)\) \((i= 1,2,\dotsc, n-2)\), \(f\in C([0,1]\times [0,\infty),[0,\infty))\), \(f(t,0)\not\equiv 0\), \(g\in C([0,1],[0,\infty))\). By using the Leray-Schauder nonlinear alternative, they prove the existence of a positive solution of \((*)\). Finally, the existence of three solutions of \((*)\) is established by means of the Leggett-Williams fixed point theorem.
scientific article; zbMATH DE number 6116528