Fractional double phase Robin problem involving variable‐order exponents and logarithm‐type nonlinearity

Abstract

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.

On a new fractional Sobolev space with variable exponent on complete manifolds

“…In particular, we shall investigate the existence of solutions to problem (1.1) using the topological degree method. This work generalizes previous results [1,2,8,9,12,24,26,27]. However, the main difficulty is presented by the fact that the p(•, •)-Laplacian operator has a more complicated nonlinearity than the p-Laplacian operator.…”