Double phase inequalities with convolution nonlinearity in exterior domains

In this paper, we discuss the existence of [Formula: see text]-solutions for two related double phase inequalities:

[Formula: see text]

where [Formula: see text] is the [Formula: see text]-Laplace operator, [Formula: see text] and [Formula: see text] where [Formula: see text] is a [Formula: see text] non-increasing function with some specific behavior near the origin. In the above context, the general form of [Formula: see text] includes the case of [Formula: see text]-Laplace and [Formula: see text]-mean curvature operator. Our study reveals a sharp distinction between [Formula: see text] and [Formula: see text]. Precisely, we show that the inequality [Formula: see text] has solutions for all [Formula: see text] and [Formula: see text]. In contrast, [Formula: see text] has solutions if and only if [Formula: see text] and [Formula: see text] are sufficiently large. We also link the solvability of [Formula: see text] with that of the corresponding equation [Formula: see text] in [Formula: see text], for which we derive optimal conditions in terms of [Formula: see text] and [Formula: see text]. The approach combines integral estimates with a new sub and supersolution method that accounts for the presence of the convolution term.