Double phase problems with competing potentials and asymptotically linear reaction

In this paper, we are concerned with the singularly perturbed double phase problem: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text] with [Formula: see text] is the [Formula: see text]-Laplacian operator, [Formula: see text] and [Formula: see text] possess global extremum points, and [Formula: see text] is asymptotically [Formula: see text]-linear at infinity. We establish the existence, nonexistence, multiplicity and concentration of solutions. Due to the appearance of asymptotically [Formula: see text]-linear nonlinearities, the method of Nehari manifold dependent on an auxiliary set is developed. In addition, refined analysis techniques and global compactness arguments are introduced to analyze the asymptotic behavior of solutions. This paper completes the results of Zhang, Zhang and Rădulescu [Math. Z. 301 (2022) 4037–4078], where [Formula: see text] is [Formula: see text]-superlinear at infinity.