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

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.