Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids

Abstract

Many electrorheological fluids are suspensions of solid particles that are exposed to a strong electric field. This causes a dramatic increase of their effective viscosity. In this paper we are concerned with a mathematical problem that is related with this non-Newtonian behavior. More precisely, we study the nonlinear stationary equation −div(|∇u|p(x)−2∇u)+|u|p(x)−2u=f(x,u) in Ω, under Dirichlet boundary conditions, where Ω is a smooth bounded domain in Rn, p>1 is a continuous function, and f(x,u) has a sublinear growth near the origin. Under various natural assumptions, by using the Morse theory in combination with local linking arguments, we obtain the existence of nontrivial weak solutions.