Stationary solutions and connecting orbits for $p$-Laplace equation

TL;DR

This paper investigates stationary solutions and connecting orbits for a one-dimensional p-Laplace equation, establishing existence results using topological methods and providing stronger conclusions than previous fixed point approaches.

Contribution

It introduces new existence results for stationary solutions and connecting orbits of the p-Laplace equation using Conley type homotopy index and continuation methods.

Findings

Existence of nontrivial stationary solutions under eigenvalue conditions.

Existence of connecting orbits between stationary solutions.

Stronger results than fixed point index methods.

Abstract

We deal with one dimensional $p$-Laplace equation of the form $$ u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, $$ under Dirichlet boundary condition, where $p>2$ and $f\colon [0,l]\times \mathbb{R}\to \mathbb{R}$ is a continuous function with $f(x,0)=0$. We will prove that if there is at least one eigenvalue of the $p$-Laplace operator between $\lim_{u\to 0} f(x,u)/|u|^{p-2}u$ and $\lim_{|u|\to +\infty} f(x,u)/|u|^{p-2}u$, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are obtained by use of Conley type homotopy index and continuation along $p$ techniques. We obtain stronger results than by use of fixed point index and additionally get the existence of a connecting orbit.