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

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, $${\mathrm{\Delta }}_{p(x)}u:=div\left(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p(x)-2}\mathrm{\nabla }u\right)=f(x)\text{quadin}\mathrm{\Omega }$$ for which we prove Harnack inequality when $f\in L^{q_0}(\mathrm{\Omega })$ if $\max \{1,\frac{N}{p_{min}}\}<q_0\le \mathrm{\infty }$. The constant in Harnack inequality depends on $u$ only through $\mathrm{\parallel }\mathrm{\mid }u{\mathrm{\mid }}^{p(x)}{\mathrm{\parallel }}_{L^1(\mathrm{\Omega })}^{p_{max}-p_{min}}$. 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 $f\in L^{q_0(x)}(\mathrm{\Omega })$ with $q_0\in C(\mathrm{\Omega })$ and $\max{1,\frac N{p(x)}}<q_0(x)$in$\Omega$. These results are then generalized to elliptic equations $$divA(x,u,\mathrm{\nabla }u)=B(x,u,\mathrm{\nabla }u)$$ with $p(x)-$type growth.