Stationary Solutions and Connecting Orbits for p-Laplace Equation

We deal with one dimensional p-Laplace equation of the form u(t) = (vertical bar u(x)vertical bar(p-2)u(x))(x) + f(x, u), x is an element of(0,l), t > 0, under Dirichlet boundary condition, where and is a continuous function with . We will prove that if there is at least one eigenvalue of the p-Laplace operator between and , then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.