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arXiv · 2507.01748

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

Lifeng Yin·Xiaoqi Liu·Yongyong Li
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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.

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Lifeng Yin, Xiaoqi Liu, Yongyong Li. 2025-07-02. Existence and concentration of nontrivial solutions for quasilinear Schr\"{o}dinger equation with indefinite potential. https://arxiv.org/abs/2507.01748

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