On the variety of solutions of 1-dimensional nonlinear eigenvalue problems

TL;DR

This work studies the 1D nonlinear eigenvalue problem $u''+\lambda f(u)=0$ on $(0,L)$ under Robin/Dirichlet boundary conditions, revealing a finite solvability threshold $\lambda^*$ and a rich tapestry of solution shapes. The authors develop a phase-plane framework, recasting solutions as trajectories on curves $\mathcal{K}_C$ and analyzing boundary-induced intersections with lines $\mathcal{L}^\pm$ to classify symmetric and asymmetric, monotone and non-monotone solutions. They prove existence/multiplicity results (including a maximal length $L_{\max}(\lambda,\alpha)$ for positive $\alpha$ and the emergence of asymmetric branches for negative $\alpha$), and show how solution counts depend on parameter regimes. Numerical continuation via pseudo-arc length confirms the analytical picture, illustrating Bratu–Gelfand-type problems and several non-convex nonlinearities, with multiple solution branches and turning points that have practical relevance for stability and bifurcation analysis.

Abstract

Second order nonlinear eigenvalue problems are considered for which the spectrum is an interval. The boundary conditions are of Robin and Dirichlet type. The shape and the number of solutions are discussed by means of a phase plane analysis. A new type of asymmetric solutions are discovered. Some numerical illustrations are given.

Figures (16)

• Figure 1: Type of solutions for $\alpha > 0$ and $\alpha < 0$.
• Figure 2: $\mathcal{K}_{C}$ for different values of $C$.
• Figure 3: Phase plane.
• Figure 4: $\lambda(C)$ for two different values of $\alpha$.
• Figure 5: Phase plane

Theorems & Definitions (39)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3