Existence of Sign-Changing Solutions to Equations Involving the One-Dimensional p-Laplacian

Abstract

We consider the equations involving the one-dimensional $p$-Laplacian $(P)\text{:}$ ${({\left|u^{\mathrm{\prime }}\left(t\right)\right|}^{p-2}{u}^{\mathrm{\prime }}(t))}^{\mathrm{\prime }}+\lambda f(u(t))=0$, and $u(0)=u(1)=0,$ where $p>1,\lambda >0,f\in C^1(\mathbb{R};\mathbb{R}),f(s)s>0,$ and $s\ne 0$. We show the existence of sign-changing solutions under the assumptions $f_{\mathrm{\infty }}={\mathrm{\text{l}}\mathrm{\text{i}}\mathrm{\text{m}}}_{|s|\to \mathrm{\infty }}(f\left(s\right)/s^{p-1})=+\mathrm{\infty }$ and $f_0={\mathrm{\text{l}}\mathrm{\text{i}}\mathrm{\text{m}}}_{|s|\to 0}(f(s)/s^{p-1})\in [0,\mathrm{\infty }]$. We also show that $(P)$ has exactly one solution having specified nodal properties for $\lambda \in (0,{\lambda }^{\mathrm{*}})$ for some ${\lambda }^{\mathrm{*}}\in (0,\mathrm{\infty })$. Our main results are based on quadrature method.