Exact number of solutions for a Neumann problem involving the p-Laplacian

Abstract

We study the exact number of solutions of the quasilinear Neumann boundary-value problem $$\displaylines{ (\varphi_p(u'(t)))'+g(u(t))=h(t)\quad\text{in } (a,b),\cr u'(a)=u'(b)=0, }$$ where $\varphi_p(s)=|s|^{p-2}s$ 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.

Keywords

• Neumann boundary value problem
• p-Laplacian
• lower-upper solutions
• exact multiplicity