Generating loops and isolas in semilinear elliptic BVP's

TL;DR

This paper analyzes the global solution structure of a semilinear elliptic boundary value problem, revealing the existence of loops and isolas in the solution set depending on parameters, and uses it as a toy model for reaction-diffusion equations.

Contribution

It characterizes the bifurcation structure of solutions, including loops and isolas, for a class of semilinear elliptic problems with parameters, extending understanding of solution topology.

Findings

Solution bifurcation diagrams depend on parameters d and q.

Existence of loops and isolas in the solution set.

Parameter λ influences both convection strength and nonlinear amplitude.

Abstract

In this paper, we ascertain the global $\lambda$-structure of the set of positive and negative solutions bifurcating from $u=0$ for the semilinear elliptic BVP \begin{equation*} \left\{\begin{array}{ll} -d\Delta u= \lambda\langle \mathfrak{a},\nabla u\rangle+u+\lambda u^{2}-u^{q} & \text{ in } \Omega, \\ u=0 & \text{ on } \partial\Omega, \end{array}\right. \end{equation*} according to the values of $d>0$ and the integer number $q\geq 4$. Figures 1.1-1.3 summarize the main findings of this paper according to the values of $d$ and $q$. Note that the role played by the parameter $\lambda$ in this model is very special, because, besides measuring the strength of the convection, it quantifies the amplitude of the nonlinear term $\lambda u^2$. We regard to this problem as a mathematical toy to generate solution loops and isolas in Reaction Diffusion equations.