Exact number of solutions for a Neumann problem involving the p-Laplacian - Justino Sanchez; Vicente Vergara (Electronic Journal of Differential Equations, 2014)
We study the exact number of solutions of the quasilinear Neumann boundary-value problem $$\displaylines{ (φ_p(u'(t)))'+g(u(t))=h(t)\quad\text{in } (a,b),\cr u'(a)=u'(b)=0, }$$ where $φ_p(s)=|s|^{p-2}s$ denotes the one-dimensional p-Laplacian. Under appropriate hypotheses on g and h, we obtain existence, multiplicity, exactness and non existence results. The existence of solutions is proved using the method of upper and lower solutions.