A concave–convex Kirchhoff type elliptic equation involving the fractionalp-Laplacian and steep well potential

AbstractIn this paper, we study the following Schrödinger–Kirchhoff type equation: (a+b∬R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−Δ)psu+λV(x)|u|p−2u=h(x)|u|r−2u+g(x)|u|q−2u, x∈RN, where s∈(0,1), 2≤p<∞, N>ps, 10, b≥0 are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution ub,λ+ by a truncated functional and discuss their asymptotical behavior for λ→+∞ when p0, we obtain a positive energy solution uλ+ and discuss their asymptotical behavior for λ→+∞ when 2p≤r