Existence and multiplicity results for elliptic problems with [formula omitted]—Growth conditions

Abstract

The variable exponent spaces are essential in the study of certain nonhomogeneous materials. In the framework of these spaces, we are concerned with a nonlinear elliptic problem involving a p(⋅)-Laplace-type operator on a bounded domain Ω⊂RN(N≥2) of smooth boundary ∂Ω. We introduce the variable exponent Sobolev space of the functions that are constant on the boundary and we show that it is a separable and reflexive Banach space. This is the space where we search for weak solutions to our equation −div(a(x,∇u))+|u|p(x)−2u=λf(x,u), provided that λ≥0 and a:Ω¯×RN→RN,f:Ω×R→R are fulfilling appropriate conditions. We use different types of mountain pass theorems, a classical Weierstrass type theorem and several three critical points theorems to establish existence and multiplicity results under different hypotheses. We treat separately the case when f has a p(⋅)−1—superlinear growth at infinity and the case when f has a p(⋅)−1—sublinear growth at infinity.