Ground State Solutions for a Non-Local Type Problem in Fractional Orlicz Sobolev Spaces

• Published in: Axioms Vol. 13; no. 5; p. 294
• Main Authors: Wang, Liben, Zhang, Xingyong, Liu, Cuiling
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2024
• Subjects: critical point; fractional Orlicz–Sobolev spaces; fractional Φ-Laplacian; Ground state; Mathematical research; Operators (mathematics); Phase transitions; Sobolev space; China
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13050294

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (−ΔΦ)su+V(x)a(|u|)u=f(x,u), x∈RN, where (−ΔΦ)s(s∈(0,1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ-Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V(x). The proof is based on a variant version of the mountain pass theorem and a Lions’ type result in fractional Orlicz–Sobolev spaces.