Global solution curves for self-similar equations

TL;DR

This paper analyzes positive solutions of certain semilinear equations on a unit ball, providing a comprehensive parameterization of solution curves, revealing solution multiplicities, Morse indices, and extending classical results to new equations including MEMS and Henon's.

Contribution

It introduces a unified approach to parameterize entire solution curves for self-similar equations, enabling detailed analysis of solution multiplicity and Morse indices, including for generalized Gelfand and MEMS equations.

Findings

Classical results on the Gelfand problem are recovered.

Infinitely many solution turns occur for the generalized Gelfand problem in dimensions n ≥ 3.

Morse index increases by one at each turn in MEMS solutions.

Abstract

We consider positive solutions of a semilinear Dirichlet problem \[ \Delta u+\lambda f(u)=0, \;\; \mbox{for $|x|<1$}, \;\; u=0 , \;\; \mbox{when $|x|=1$} \] on a unit ball in $R^n$. For four classes of self-similar equations it is possible to parameterize the entire (global) solution curve through the solution of a single initial value problem. This allows us to derive results on the multiplicity of solutions, and on their Morse indices. In particular, we easily recover the classical results of D.D. Joseph and T.S. Lundgren [6] on the Gelfand problem. Surprisingly, the situation turns out to be different for the generalized Gelfand problem, where infinitely many turns are possible for any space dimension $n \geq 3$. We also derive detailed results for the equation modeling electrostatic micro-electromechanical systems (MEMS), in particular we easily recover the main result of Z. Guo and J. Wei [4], and we show that the Morse index of the solutions increases by one at each turn. We also consider the self-similar Henon's equation.