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 $f:[0,l]\times \mathbb{R}\to \mathbb{R}$ 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.