Hölder continuity of weak solution to a nonlinear problem with non-standard growth conditions

Abstract

Abstract We study the Hölder continuity of weak solution u to an equation arising in the stationary motion of electrorheological fluids. To this end, we first obtain higher integrability of Du in our case by establishing a reverse Hölder inequality. Next, based on the result of locally higher integrability of Du and difference quotient argument, we derive a Nikolskii type inequality; then in view of the fractional Sobolev embedding theorem and a bootstrap argument we obtain the main result. Here, the analysis and the existence theory of a weak solution to our equation are similar to the weak solution which has been established in the literature with 3dd+2≤p∞≤p(x)≤p0<∞ $\frac{3d}{d+2}\leq p_{\infty}\leq p(x)\leq p_{0}<\infty$, and in this paper we confine ourselves to considering p(x)∈(3dd+2,2) $p(x)\in(\frac{3d}{d+2},2)$ and space dimension d=2,3 $d=2,3$.

Keywords

• Higher integrability
• Hölder continuity
• Nonlinear problem
• Fractional Sobolev space
• Electrorheological fluids