Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations

Abstract

Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where ε , λ > 0 \varepsilon ,\lambda \gt 0 are parameters, N ⩾ 2 N\geqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , I α {I}_{\alpha } is the Riesz potential with 0 < α < N 0\lt \alpha \lt N , and V ∈ C ( R N , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{N},{\mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.