Three positive solutions of the one-dimensional generalized Hénon equation (Q2258462)

The paper is concerned with the existence of at least three positive solutions for the one-dimensional generalized Hénon equation: \[ u''+h(x)u=0,\;u>0\;\text{ in }\,(a,b) \] subject to the Dirichlet boundary conditions \(u(a)=u(b)=0\). Here, \(p>1\) and \(h\in L^\infty(a,b)\) is a positive function. The Mountain Pass theorem yields the existence of at least one positive solution. When \(h\) is even and identical to \(0\) on a ``large'' sub-interval of \((a,b)\), it is well known that the problem has at least three positive solutions. The main purpose of the paper is to remove the evenness from the assumption on \(h\) while keeping the hypothesis that \(h\) vanishes on some sub-interval of \((a,b)\) and has some behavior near the endpoints \(a, b\). The proof uses variational techniques.