Infinitely many solutions for three classes of self-similar equations, with the $p$-Laplace operator

TL;DR

This paper proves the existence of infinitely many positive solutions for three classes of self-similar equations involving the p-Laplace operator, extending classical results from the case p=2 to general p values.

Contribution

It generalizes known results for p=2 to the p-Laplace operator case and extends classical solution curve analyses to a broader class of nonlinear equations.

Findings

Existence of infinitely many positive solutions for the equations.

Extension of classical results to the p-Laplace operator case.

Generalization of MEMS modeling results.

Abstract

We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand equation, the equation modeling electrostatic micro-electromechanical systems (MEMS), and a polynomial nonlinearity. We extend the classical results of D.D. Joseph and T.S. Lundgren \cite{JL} to the case $p \ne 2$, and we generalize the main result of Z. Guo and J. Wei \cite{GW} on the equation modeling MEMS.