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

Abstract

The paper deals with the logarithmic fractional equations with variable exponents where and denote the variable ‐order ‐fractional Laplace operator and the nonlocal normal ‐derivative of ‐order, respectively, with and ( ) being continuous. Here, is a bounded smooth domain with ( ) for any and are a positive parameters, and are two continuous functions, while variable exponent can be close to the critical exponent , given with and for . Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, , for any . In the second case, we study the critical exponent, namely, for some . Then, using variational methods, we prove the existence and multiplicity of solutions and existence of ground state solutions to the above problem.