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

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.