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

• Published in: Boundary value problems Vol. 2018; no. 1; pp. 1 - 23
• Main Authors: Tan, Zhong, Zhou, Jianfeng, Zheng, Wenxuan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 31.08.2018; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Continuity (mathematics); Difference and Functional Equations; Electrorheological fluids; Fractional Sobolev space; Higher integrability; Hölder continuity; Integral calculus; Mathematics; Mathematics and Statistics; Nonlinear problem; Ordinary Differential Equations; Partial Differential Equations; Higher integrability; Hölder continuity; Electrorheological fluids; Fractional Sobolev space; Nonlinear problem
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-1051-6

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 3 d d + 2 ≤ p ∞ ≤ p ( x ) ≤ p 0 < ∞ , and in this paper we confine ourselves to considering p ( x ) ∈ ( 3 d d + 2 , 2 ) and space dimension d = 2 , 3 .