Abstract
This paper is devoted to the study of the L ∞-bound of solutions to a double-phase problem with concave-convex nonlinearities by applying the De Giorgi's iteration method and the localization method. Employing this and a variant of Ekeland's variational principle, we provide the existence of at least two distinct nontrivial solutions belonging to L ∞-space when the convex term does not satisfy the Ambrosetti-Rabinowitz condition in general. In addition, our problem has a sequence of multiple small energy solutions whose L ∞-norms converge to zero. To achieve this result, we utilize the modified functional method and the dual fountain theorem as the main tools.
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