Infinitely many solutions for problems in fractional Orlicz–Sobolev spaces

Abstract

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger Φ-Laplace equation ( − Δ ) Φ u + V ( x ) φ ( u ) = ξ ( x ) f ( u ) in ℝ d , where Φ(t)= ∫ 0tφ(s)ds is an N-function, ΔΦ is the Φ-Laplacian operator, V:ℝd→ℝ is a continuous function, ξ is a function with sign-changing on ℝd and the nonlinearity f is sublinear as |u|→∞. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces. We also study the existence and multiplicity of solutions to the general fractional Φ-Laplacian equations of Kirchhoff type M ( ∫ ℝ 2 d Φ ( u ( x ) − u ( y ) K ( | x − y | ) ) d x d y N ( | x − y | ) ) ( − Δ ) Φ K , N u = f ( x , u ) in Ω , u = 0 in ℝ d ∖ Ω , where Ω is an open bounded subset of ℝd with smooth boundary ∂Ω, d>2, and M:ℝ0+→ℝ+ is a continuous function and f:Ω×ℝ→ℝ is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory.