Multiplicity results for variable-order fractional Laplacian equations with variable growth

In this paper, we study the multiplicity of solutions for an elliptic type problem driven by the variable-order fractional Laplace operator involving variable exponents. More precisely, we consider {(-Delta)(s(.))u + lambda V(x)u = alpha vertical bar u vertical bar(p(x)-2)u + beta vertical bar u vertical bar(q(x)-2)u in Omega, u = 0 in R-N\Omega, where N >= 1, s(.) : R-N x R-N -> (0, 1) is a continuous function, Omega is a bounded domain in R-N with N > 2s(x,y) for all (x, y) is an element of Omega x Omega, (-Delta)(s(.)) is the variable-order fractional Laplace operator, lambda > 0 is a parameter, V : Omega -> [0, infinity) is a continuous function, alpha,beta > 0 are two parameters and p, q is an element of C(Omega). Under some suitable assumptions, we show that the above problem admits at least two distinct solutions by applying the mountain pass theorem and Ekeland's variational principle. Then we prove that these two solutions converge to two solutions of a limit problem as lambda -> infinity. Moreover, we obtain the existence of infinitely many solutions for the limit problem. (C) 2018 Elsevier Ltd. All rights reserved.