Existence of solution to singular Schrödinger systems involving the fractional p‐Laplacian with Trudinger–Moser nonlinearity in ℝN

Abstract

In this paper, we study the existence of weak solution for singular fractional Schrödinger system in involving Trudinger–Moser nonlinearity as follows: where N ≥ 1, 0 < s < 1, N = ps, γ ∈ [0, N), and H has exponential growth and does not satisfy the Ambrosetti–Rabinowitz condition. Note that our problem is the lack of compactness. First, we give a version of vanishing lemma due to Lions; using that result and a version of Mountain pass theorem without (PS) condition, we obtain the existence of nontrivial solution to the above system. When H satisfies the Ambrosetti–Rabinowitz condition, motivated by the work of Chen et al., we study the existence of nontrivial solution to singular Schrödinger system involving the fractional (p1, p)‐Laplace and Trudinger–Moser nonlinearity. In our best knowledge, this is the first time the above problems are studied.