Local bounds, Harnack inequality and H\"older continuity for divergence type elliptic equations with nonstardard growth

Abstract

In this paper 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, \[ \Delta_{p(x)}u:=\mbox{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)=f(x)\quad\mbox{in}\quad\Omega \] for which we prove Harnack inequality when $f\in L^{q_0}(\Omega)$ if $\max\{1,\frac N{p_{min}}\}<q_0\le \infty$. The constant in Harnack inequality depends on $u$ only through $\||u|^{p(x)}\|_{L^1(\Omega)}^{p_{max}-p_{min}}$. 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 $f\in L^{q_0(x)}(\Omega)$ with $q_0\in C(\Omega)$ and $\max\{1,\frac N{p(x)}\}<q_0(x)$ in $\Omega$. These results are then generalized to elliptic equations \[ \mbox{div}A(x,u,\nabla u)=B(x,u,\nabla u) \] with $p(x)-$type growth.