Quasilinear Schrödinger equations with general sublinear conditions

Abstract

In this paper, we study the quasilinear Schrödinger equations $$-\Delta u+V(x)u+\Delta(u^2)u = f(x, u),\qquad\forall x\in\mathbb{R}^N,$$ where $V\in C(\mathbb{R}^N;\mathbb{R})$ may change sign and $f$ is only locally defined for $|u|$ small. Under some new assumptions on $V$ and $f$, we show that the above equation has a sequence of solutions converging to zero. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.

Keywords

• variational methods
• critical points
• quasilinear schrödinger equations