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

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