Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces

In this study, we examine a particular type of fractional (phi(1), phi(2))-Laplacian problem, which can be represented by the following equation: {(-Delta)(phi 1)(s)((center dot)) u + (-Delta)(phi 2)(s)(center dot) u + V(epsilon x)(phi(1)(vertical bar u vertical bar)u + phi(2)(vertical bar u vertical bar)u) = f(x, u) in R-n, u is an element of W-epsilon, u > 0 in R-n. Here, s is an element of (0, 1), epsilon > 0, and (-Delta)(phi i)(s)((center dot)) (where i = 1, 2) is a fractional phi(i)-Laplacian operator. The potential function V : R-n -> R is a continuous, possibly unbounded, function and f : R -> R is a continuous nonlinearity with Orlicz subcritical growth. Our goal is to investigate the multiplicity and concentration properties of the solutions to this problem as epsilon approaches zero, using the Nehari manifold and the penalization method as our primary tools.