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arXiv · 2009.08531

Positive Solutions For a Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$

Bruno Mascaro·Gaetano Siciliano
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Abstract

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 $V$.

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Bruno Mascaro, Gaetano Siciliano. 2022-12-12. Positive Solutions For a Schrödinger-Bopp-Podolsky system in $\mathbb R^{3}$. https://doi.org/10.46298/cm.10363

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