Partial Regularity of Minimizers of Asymptotically Convex Functionals with Morrey Coefficients

Abstract We consider minimizers of the functional ∫ Ω f ( x , u , D u ) d x , where f is asymptotically related the function ( x , u , ξ ) ↦ a ( x , u ) G ( ξ ) , with G a function with p-Uhlenbeck structure and a ∈ C 0 ( Ω × R N ) . The main contribution of this article is to refine the growth estimate imposed on f. In particular, we assume that | f ( x , u , ξ ) | ≤ μ 1 ( x ) + μ 2 ( x ) | u | s + M ( 1 + | ξ | 2 ) p 2 , where μ 2 ∈ L γ , β ( Ω ) for some numbers β and γ. By assuming only that the coefficient μ 2 belongs to a Morrey space, relative to existing results in the literature we are able to better match our results to a given problem.