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

N. WOLANSKI

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2015 vol. 56 p. 73 - 105

0041-6932

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}ig(|abla u|^{p(x)-2}abla uig)=f(x)quadmbox{in}quadOmega]for which we prove Harnack´s inequality when $fin L^{q_0}(Omega)$ if $max{1,rac N{p_1}}<q_0le infty$. The  constant in Harnack´s inequality depends on $u$ only through  $||u|^{p(x)}|_{L^1(Omega)}^{p_2-p_1}$. 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 $fin L^{q_0(x)}(Omega)$ with $q_0in C(Omega)$ and $max{1,rac N{p(x)}}<q_0(x)$ in $Omega$.These results are then generalized to elliptic equations[mbox{div}A(x,u,abla u)=B(x,u,abla u)]with $p(x)-$type growth.