On a singular one-dimensional \(p\)-Laplacian-like equation with Neumann boundary conditions (Q2717971)

Sufficient and necessary conditions for the solvability of a boundary value problem for the equation NEWLINE\[NEWLINE(A(|u'|)u')''-g(u(t))=h(t), \quad 0<t<1,NEWLINE\]NEWLINE with Neumann conditions \(u'(0)=u'(1)=0\) are established for \(h(t) \in L(0,1)\). Here, \(A(x)>0\) for \(x>0\) and \(g(x)\) is continuous everywhere except at the point of singularity \(x=0\). Under some additional assumptions for \(A, g\) the authors obtain an a priori estimate on a solution by standard tools and prove a theorem on existence. The authors demonstrate that the conditions imposed are sharp. For this they show a nonexistence result.