ABSTRACT

This paper is motivated by the study of the following quasilinear Schrödinger equation $-\mathrm{\Delta }u+V\left(x\right)u-\left[\mathrm{\Delta }\right(1+u^2{)}^{\frac{1}{2}}]\frac{u}{2(1+u^2{)}^{\frac{1}{2}}}=\lambda h(u),x\in {\mathbb{R}}^N,$where $N\ge 3$, $\lambda >0$ is a parameter and $V\left(x\right)$ is a given positive potential. As an example, the nonlinearity includes the pure power type of $h\left(u\right)=|u{|}^{p-2}u$ for the well-studied case $12-4\sqrt{6}<p<2^{\ast }$, and the case $2<p<12-4\sqrt{6}$ in which few existence results are known. Distinguishing from the existing results in the literature, we are more interested in the existence and multiplicity of sign-changing solutions for the above problem.

中文翻译：

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拟线性薛定谔方程的变号孤波的存在性和多重性

摘要

本文的动机是研究以下拟线性薛定谔方程$-\mathrm{\Delta }你+五\left(X\right)你-\left[\mathrm{\Delta }\right(1+{你}^2{)}^{\frac{1}{2}}]\frac{你}{2(1+{你}^2{)}^{\frac{1}{2}}}=\lambda H(你),X\in {\mathbb{R}}^{ñ},$在哪里$ñ\ge 3$,$\lambda >0$是一个参数并且$五\left(X\right)$是给定的正电位。例如，非线性包括纯功率类型$H\left(你\right)=|你{|}^{p-2}你$对于经过充分研究的案例$12-4\sqrt{6}<p<2^{*}$, 情况$2<p<12-4\sqrt{6}$其中已知的存在结果很少。与文献中的现有结果不同，我们对上述问题的符号转换解决方案的存在性和多样性更感兴趣。