Existence of S-shaped type bifurcation curve with dual cusp catastrophe via variational methods

TL;DR

This paper proves the existence of an S-shaped bifurcation curve with dual cusp catastrophe for a class of p-Laplacian equations, revealing complex solution structures and stability properties using variational methods.

Contribution

It introduces a nonlinear generalized Rayleigh quotient method to analyze multiple positive solutions and bifurcation phenomena in non-Lipschitz p-Laplacian problems.

Findings

Existence of multiple positive solutions with stability analysis.

Identification of S-shaped bifurcation curve with dual cusp catastrophe.

Results are new even for one-dimensional, p=2 cases.

Abstract

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form $$ -\Delta_p u= f(\mu,\lambda, u)~ \mbox{in} ~\Omega \subset \mathbb{R}^N $$ where $\Delta_p$ is a $p$-Laplacian, $p>1$, $N\geq 1$, $\mu, \lambda \in \mathbb{R}$. We deal with relatively unexplored cases when $f(\mu,\lambda, u)$ is non-Lipschitz at $u=0$, $f(\mu,\lambda, 0) = 0 $ and $ f(\mu,\lambda, u) <0$, $u \in (0,r)$, for some $r<+\infty$. We develop the nonlinear generalized Rayleigh quotients method to find a range of parameters where the equation may have distinct branches of positive solutions. As a consequence, applying the Nehari manifold method and the mountain pass theorem, we prove that the equation for some range of values $\mu, \lambda$, has at least three positive solutions with two linearly unstable solutions and one linearly stable. The results evidence that the bifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophe which is characterized by the fact that the corresponding dynamic equation has stable states only within the cusp-shaped region in the control plane of parameters. Our results are new even in the one-dimensional case and $p=2$.