论文标题
$ \ mathbb r^{3} $中的Schrödinger-Bopp-Podolsky系统的积极解决方案
Positive Solutions For a Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$
论文作者
论文摘要
我们考虑以下schrödinger-bopp-podolsky系统在$ \ mathbb r^{3} $中 $$ \ left \ {\ begin {array} {c} - \ varepsilon^{2}Δu + v（x）u + ϕu = f（u）\\ - \\ - \ varepsilon^{2}Δ u^{2} \\ \ end {array} \ right。$$ 其中$ \ varepsilon> 0 $ with $ v：\ mathbb {r}^{3} \ rightarrow \ mathbb {r}，f：\ mathbb {r} \ rightArrow \ rightArrow \ mathbb {r} $满足合适的假设。通过使用变分方法，我们证明了以下$ M $的LJusternick-Schnirmann类别（潜在的$ V $的最小值）估算了积极解决方案的数量。
We consider the following Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$ $$\left\{ \begin{array}{c} -\varepsilon^{2} Δu + V(x)u + ϕu = f(u)\\ -\varepsilon^{2} Δϕ+ \varepsilon^{4} Δ^{2}ϕ= 4π\varepsilon u^{2}\\ \end{array} \right.$$ where $\varepsilon > 0$ with $ V:\mathbb{R}^{3} \rightarrow \mathbb{R}, f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy suitable assumptions. By using variational methods, we prove that the number of positive solutions is estimated below by the Ljusternick-Schnirelmann category of $M$, the set of minima of the potential $V$.
