Two positive solutions for second-order quasilinear differential equation boundary value problems with sign changing nonlinearities
The authors study the sovability of boundary value problems for the equation \[ \bigl(\Phi(y')\bigr)'+q(t)f(t,y)=0,\;\;0<t<1, \] with boundary conditions \[ y(0)=y(1)=0\quad\text{or}\quad y(0)=y'(1)=0, \] where \(\Phi(v)=| v|^{p-2}v, p>1,\quad q:(0,1)\to[0,\infty)\) is continuous, \(q(t)\not\equiv0\) on any subinterval of \((0,1),\) \(\int_0^1q(t)dt<\infty\), and \(f\) is allowed to change sign, i.e., \(f:[0,1]\times[0,\infty)\to[-M,\infty)\) (with any constant \(M>0\)) is continuous and such that \(f(t,0)\geq0\) \((\not\equiv0)\) for \(t\in[0,1].\) The authors have imposed growth conditions on \(f\) to obtain at least two positive solutions for the considered boundary value problems. Moreover, they have applied a new fixed-point theorem in cones.
scientific article; zbMATH DE number 2091414