Combined effects of concave and convex nonlinearities for the generalized Chern–Simons–Schrödinger systems with steep potential well and 1 < p < 2 < q < 6

Abstract

In this paper, we study the Chern–Simons–Schrödinger system with a steep potential well and 1 < p < 2 < q < 6. First, by using the truncation technique, we prove that this system possesses a positive energy solution. Second, the concentration behavior of the positive energy solutions as λ → +∞ and κ → 0 are also considered. Finally, we obtain a negative energy solution via the Ekeland variational principle.