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

Abstract

In this paper, we study the following Schrödinger–Kirchhoff type equation: , where , , N>ps, and , 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 by a truncated functional and discuss their asymptotical behavior for when p<r<2p and b lies in suitable range. Secondly, for all b>0, we obtain a positive energy solution and discuss their asymptotical behavior for when , where . Moreover, we obtain other asymptotical behavior of positive energy solution when . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.