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

TL;DR

This paper establishes conditions for the existence of sign-changing solutions to a nonlinear p-Laplacian boundary value problem, extending known results from the linear case to more general nonlinear scenarios.

Contribution

It provides new sufficient conditions for solutions with prescribed nodal properties for the nonlinear p-Laplacian, generalizing previous results for p=2.

Findings

Conditions for existence of sign-changing solutions.

Results applicable to more general nonlinear cases.

Extension of known results from linear to nonlinear p-Laplacian.

Abstract

We study the nonlinear one-dimensional $p$-Laplacian equation $$ -(y'^{(p-1)})'+(p-1)q(x)y^{(p-1)}=(p-1)w(x)f(y) on (0,1),$$ 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.