On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the m-Laplacian. (P) - M ∫ω|∇u|mdx Δ mu = f(x,u)in ω, u = 0, on ∂ω, where ω ⊂RN is a bounded domain with smooth boundary, M is a Kirchhoff function, N ≥ m ≥ 2 and f C(ω¯ × R) is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser-Trudinger inequality. Our interest then is to study (P) without the analogue of Ambrosetti-Rabinowitz superquadratic condition ((AR)γ condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and m-Laplacian cases.