Existence and multiplicity of solutions to concave–convex-type double-phase problems with variable exponent

• Published in: Nonlinear analysis: real world applications Vol. 67; p. 103627
• Main Authors: Kim, In Hyoun, Kim, Yun-Ho, Oh, Min Wook, Zeng, Shengda
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2022
• Subjects: De Giorgi iteration; Double phase operator with variable exponent; Dual fountain theorem; Modified functional methods; Musielak–Orlicz Sobolev space; Modified functional methods; Musielak–Orlicz Sobolev space; Double phase operator with variable exponent; De Giorgi iteration; Dual fountain theorem
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2022.103627

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.