  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.
