Combined Effects of Concave and Convex Nonlinearities for Kirchhoff Type Equations with Steep Potential Well and 1 < p < 2 < q < 4

Abstract

In this paper, we study a class of Kirchhoff type equations with concave and convex nonlinearities and steep potential well. Firstly, we obtain a positive energy solution $$u_{b,\lambda}^ + $$ by a truncated functional. Furthermore, the concentration behavior of $$u_{b,\lambda}^ + $$ is also explored on the set V−1 (0) as λ → ∞. Secondly, we also give the existence of a negative solution $$u_{b,\lambda}^ - $$ via Ekeland variational principle. Finally, we show a nonexistence result of the nontrivial solutions.