Local Bounds, Harnack's Inequality and Hölder Continuity for for divergence type elliptic equations with non-standard growth

Abstract

We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, ∆p(x)u := div |∇u| p(x)−2∇u = f(x) in Ω, for which we prove Harnack’s inequality when f ∈ Lq0 (Ω) if max{1, N p1 } < q0 ≤ ∞. The constant in Harnack’s inequality depends on u only through k|u| p(x)k p2−p1 L1(Ω) . Dependence of the constant on u is known to be necessary in the case of variable p(x). As in previous papers, log-H¨older continuity on the exponent p(x) is assumed. We also prove that weak solutions are locally bounded and H¨older continuous when f ∈ Lq0(x) (Ω) with q0 ∈ C(Ω) and max{1, N p(x) } < q0(x) in Ω. These results are then generalized to elliptic equations div A(x, u, ∇u) = B(x, u, ∇u) with p(x)-type growth.