Existence and concentration of nontrivial solutions for quasilinear Schrödinger equation with indefinite potential

Abstract

This paper is concerned with the quasilinear Schrödinger equation \begin{align*} -Δu+V(x)u+\frac{k}{2}Δ(u^2)u=f(u)\quad \text{in}~~\mathbb{R}^N\text{,} \end{align*} where $N\ge 3$, $k>0$, $V\in C(\mathbb{R})$ is an indefinite potential. Under structural conditions on the potential $V$ and the nonlinearity $f$, we establish the existence of a nontrivial solution through a combination of a local linking argument, Morse theory, and the Moser iteration. Moreover, if $f$ is odd, we obtain an unbounded sequence of nontrivial solutions via the symmetric Mountain Pass Theorem. Additionally, as $k\to 0$, we analyze the concentration behavior of nontrivial solutions.