Existence and concentration of nontrivial solutions for quasilinear Schr\"{o}dinger equation with indefinite potential

TL;DR

This paper proves the existence and concentration of nontrivial solutions for a quasilinear Schrödinger equation with an indefinite potential, using variational methods and topological tools, and explores solution behavior as a parameter tends to zero.

Contribution

It introduces new existence results for solutions under indefinite potentials and analyzes their concentration behavior as a parameter approaches zero.

Findings

Existence of nontrivial solutions established.

Multiple solutions obtained when nonlinearity is odd.

Solutions concentrate as the parameter k approaches zero.

Abstract

This paper is concerned with the quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u+V(x)u+\frac{k}{2}\Delta(u^2)u=f(u)\quad \text{in}~~\mathbb{R}^N\text{,} \end{align*} where $N\geq 3$, $k>0$, $V\in C(\R)$ 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 $k\rightarrow0$, we analyze the concentration behavior of nontrivial solutions.