Abstract
We investigate the multiplicity of solutions for problems involving the fractional N-Laplacian. We obtain three theorems depending on the source terms in which the nonlinearities cross some eigenvalues. We obtain these results by direct computations with the eigenvalues and the corresponding eigenfunctions for the fractional N-Laplacian eigenvalue problem in the fractional Orlicz–Sobolev spaces, the contraction mapping principle on the fractional Orlicz–Sobolev spaces and Leray–Schauder degree theory.
Highlights
In this paper we consider the existence and multiplicity of solutions in W0sLG( ) ∩ C( ) for the following fractional N-Laplacian Dirichlet boundary value problems with jumping nonlinearties; (–)sg u(x) = b(s,g,α)g u(x) u(x)+ |u(x)|a(s,g,α)g u(x) u(x)– |u(x)|+ τg φ1(s,g,α)(x) φ1(s,g,α) |φ1(s,g,α) (x) (x)| (1.1)G u(x) dx = α, in, u(x) = 0 on ∂, where is a bounded domain of RN with a smooth boundary ∂, N ≥ 1, 0 < s < 1, α > 0, g is an odd, strictly increasing continuous function from [0, ∞) onto [0, ∞) with g(0) = 0 and limt→∞ g(t) = ∞, G(t) = |t| 0 g(τ ) dτ, t
The fractional Sobolev spaces with variable exponent are defined by
Results on the fractional Sobolev spaces with variable exponent and the corresponding nonlocal equations were obtained in [14]
Summary
Let g be an odd, strictly increasing continuous function from [0, ∞) onto [0, ∞), g(0) = 0 and limt→∞ g(t) = ∞. Our main theorems are as follows: Theorem 1.1 Let 0 < s < 1, sg0 < N , α > 0 and g be an odd, strictly increasing continuous function. (iii) There exists τ1(s,g,α) < 0 such that for any τ with τ1(s,g,α) ≤ τ < 0, (1.1) has exactly two solutions in W0sLG( ) ∩ C( ).
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