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

The paper deals with the logarithmic fractional equations with variable exponents {(-Delta)(p1(.))(s1(.)) (u)+(-Delta)(p2(.))(s2(.) )(u)+vertical bar u vertical bar((p) over bar1(x)-2)u+vertical bar u vertical bar((p) over bar2(x)-2)u=lambda b(x)vertical bar u vertical bar(alpha(x)-2)u( ) +mu a(x)vertical bar u vertical bar(r(x)-2) u log vertical bar u vertical bar + mu c(x) vertical bar u vertical bar(eta(x)-2)u, x is an element of Omega, N-p1(.)(s1(.))(u) + N-p2(.)(s2(.))(u) + beta(x)(vertical bar u vertical bar(p) over bar1(x)-2 + vertical bar u vertical bar((p) over bar2(x)-2)u) = 0, x is an element of R-N\(Omega) over bar, where (-Delta)(pi(.))(si(.)) and N-pi(.)(si(.)) denote the variable s(i)(.) -order p(i)(.) -fractional Laplace operator and the nonlocal normal p(i)(.) -derivative of s(i)(.) -order, respectively, with s(i)(.) : R-2N -> (0,1) and p(i)(.) : R-2N -> (1,infinity) (i is an element of {1,2}) being continuous. Here, Omega subset of R-N is a bounded smooth domain with N > p(i)(x,y)s(i)(x,y) ( i is an element of {1,2} ) for any (x,y) is an element of (Omega) over barx (Omega) over bar,lambda and mu are a positive parameters, r(.) and eta(.) are two continuous functions, while variable exponent alpha(x) can be close to the critical exponent p(2s2*)(x) = N (p) over bar (2)(x)/(N - (s) over bar (2)(x)(p) over bar (2)(x)) , given with (p) over bar (2)(x) = p(2)(x,x) and ( s) over bar (2)(x)=s(2)(x,x) for x is an element of (Omega) over bar. Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, alpha(x) < p(2s)(2)*(x) , for any x is an element of (Omega) over bar . In the second case, we study the critical exponent, namely, alpha(x) = p(2s)(2)*(x) for some x is an element of (Omega) over bar. Then, using variational methods, we prove the existence and multiplicity of solutions and existence of ground state solutions to the above problem.