论文标题
带有Stein-Weiss类型卷积和关键指数非线性的Quasilinearschrödinger方程
Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$
论文作者
论文摘要
在本文中，我们调查了以下类别{schrödinger}方程的积极解决方案的存在，涉及Stein-Weiss类型卷积 \ begin {align*} -Δ_nu-Δ_n（u^{2}）u +v（x）| u |^{n -2} u = \ left（\ int _ {\ mathbb r^n} \ frac {f（y，u）} {| y | y |^β| x--- | x-y |^μ} 〜d dy \ right） \ text {in} \; \ Mathbb r^n， \end{align*} where $N\geq 2,\,$ $0<μ<N,\, β\geq 0,$ and $2β+μ\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb r^n $和一些适当的假设。在Trudinger-Moser不平等的意义上，非线性$ f：\ Mathbb r^n \ time \ time \ to \ mathbb r \ to \ mathbb r $是一个连续的函数，具有关键的指数增长，$ f（x，s）= \ int_ {0}^s f（x，x，x，x，x，x，x，x，x，x，x，t）dt $是$ f $的原始$。
In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schrödinger} equations involving Stein-Weiss type convolution \begin{align*} -Δ_N u -Δ_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^β|x-y|^μ}~dy\right)\frac{f(x,u)}{|x|^β} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\geq 2,\,$ $0<μ<N,\, β\geq 0,$ and $2β+μ\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriate assumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of $f$.
