Existence of sign-changing solutions for the nonlinear $p$-Laplacian boundary value problem

Abstract

We study the nonlinear one-dimensional $p$-Laplacian equation $$-(y^{\prime (p-1)}{)}^{\prime }+(p-1)q(x)y^{(p-1)}=(p-1)w(x)f(y)on(0,1),$$ with linear separated boundary conditions. We give sufficient conditions for the existence of solutions with prescribed nodal properties concerning the behavior of $f(s)/s^{(p-1)}$ when $s$ are at infinity and zero. These results are more general and complementary for previous known ones for the case $p=2$ and $q$ is nonnegative.

22 pages