On the asymptotically cubic generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

Abstract

In this paper, we consider the non-existence and existence of solutions for a generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation. When the non-linearityh(u) shows critical or supercritical growth at infinity, the non-existence result for a quasilinear Schrödinger equation is proved via the Pohožaev identity. Ifh(u) shows asymptotically cubic growth at infinity, the existence of positive radial solutions for the quasilinear Schrödinger equation is obtained whenbis large or equal to 0 andbis equal to 0 by the variational methods. Moreover, some properties are established as the parameterbtends to be 0.