In this paper, we obtain Sobolev type embedding results for the fractional Sobolev spaces with variable order and variable exponents. Then we study multiplicity and regularity of the solutions for a class of nonlocal problems involving variable order fractional p(•)-Laplacian and concave-convex nonlinearities of the formwhere λ > 0, Ω is a smooth and bounded domain inand f : Ω × R → R is a Carathéodory function with subcritical growth. Using variational methods, under some suitable assumptions on s, p, α and f, we prove the existence of the multiple solutions and uniform estimate for the solutions of the above problem.
Funding informationFAPESP Thematic Project titled Systems and partial differential equations; INdAM-GNAMPA project titled Equazioni alle derivate parziali: problemi e modelliThe paper deals with the logarithmic fractional equations with variable exponentswhere (−Δ)p i (•) denote the variable s i (•)-order p i (•)-fractional Laplace operator and the nonlocal normal p i (•)-derivative of s i (•)-order, respectively, with) for any (x, 𝑦) ∈ Ω × Ω, 𝜆 and 𝜇 are a positive parameters, r(•) and 𝜂(•) are two continuous functions, while variable exponent 𝛼(x) can be close to the critical exponent p * 2s 2 (x) = Np 2 (x)∕(N − s 2 (x)p 2 (x)), given with p 2 (x) = p 2 (x, x) and s 2 (x) = s 2 (x, x) for x ∈ Ω. Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, 𝛼(x) < p * 2 s 2 (x), for any x ∈ Ω. In the second case, we study the critical exponent, namely, 𝛼(x) = p * 2 s 2 (x) for some x ∈ Ω. Then, using variational methods, we prove the existence and multiplicity of solutions and existence of ground state solutions to the above problem.
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