A New Solution Operator of One-Dimensional p -Laplacian with a Sign-Changing Weight and Its Application

Abstract

We establish a new solution operator for the following problem $-{\phi }_p(u\prime )\prime =g$, $t\in (\mathrm{0,1})$, $u(0)=0=u(1)$, where ${\phi }_p(x)=|x{|}^{p-2}x$, $p>1$. $g$ may be singular at the boundary or change signs or may not be in $L^1(\mathrm{0,1})$ so that this solution operator can cover larger class of $g$ than previously known ones. As an application, by checking complete continuity of the solution operator, we show the existence of nontrivial solutions for $p$-Laplacian ${\phi }_p(u\prime )\prime +\lambda h(t)f(u(t))=0$, $t\in (\mathrm{0,1})$, $u(0)=0=u(1)$, where $\lambda >0$ a parameter and $f\in C(ℝ,ℝ)$ and $f(0)>0$ and $h$ may change signs or may be beyond of $L^1(\mathrm{0,1})$.