MULTIPLE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO

Abstract We consider quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $-\Delta u+V(x)u-u\Delta (u^2)=g(u)$ , where the potential V is allowed to be sign-changing and the nonlinearity g is sublinear at zero. Except for being subcritical, no additional condition is imposed on $g(u)$ for $|u|$ large. We obtain a sequence of solutions with negative energy and converging to zero via Clark’s theorem. We also obtain a similar result for fourth-order quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $\Delta ^2u-\Delta u+ V(x)u-u\Delta (u^2)=g(u)$ .