Existence and multiplicity of solutions to concave–convex-type double-phase problems with variable exponent. (October 2022)

• Record Type: Journal Article
• Authors: Kim, In Hyoun; Kim, Yun-Ho; Oh, Min Wook; Zeng, Shengda
• Is Part Of: Nonlinear analysis. Volume 67(2022)
• Publication Date: 2022-10
• Subjects: Double phase operator with variable exponent -- De Giorgi iteration -- Dual fountain theorem -- Modified functional methods -- Musielak–Orlicz Sobolev space; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2022.103627
• Languages: English
• ISSNs: 1468-1218

Abstract

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.