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

DOI10.46298/CM.10363arXiv2009.08531MaRDI QIDQ6349345

Publication date: 17 September 2020

Abstract: We consider the following Schr"odinger-Bopp-Podolsky system in

m a t h b b R^{3}

left{ �egin{array}{c} -varepsilon^{2} Delta u + V(x)u + phi u = f(u)\ -varepsilon^{2} Delta phi + varepsilon^{4} Delta^{2}phi = 4pivarepsilon u^{2}\ end{array} ight.

where

v a r e p s i l o n > 0

with

V : m a t h b b R^{3} i g h t a r r o w m a t h b b R, f : m a t h b b R i g h t a r r o w m a t h b b 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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