Existence and multiplicity of solutions to concave–convex-type double-phase problems with variable exponent. (October 2022) - Kim, In Hyoun; Kim, Yun-Ho; Oh, Min Wook; Zeng, Shengda
Abstract: This paper is devoted to the study of the L ∞ -bound of solutions to the double-phase nonlinear problem with variable exponent by the case of a combined effect of concave–convex nonlinearities. The main tools are the De Giorgi iteration method and a truncated energy technique. Applying this and a variant of Ekeland's variational principle, we give the existence of at least two distinct nontrivial solutions belonging to L ∞ -space when the condition on a nonlinear convex term does not assume the Ambrosetti–Rabinowitz condition in general. In addition, our problem admits a sequence of small energy solutions whose converge to zero in L ∞ space. To achieve this result, we apply the modified functional method and global variational formulation as the main tools.