Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory

Abstract

This paper concerns the following nonlinear Choquard equation: \begin{document}$ \begin{equation} -\varepsilon^{2}\Delta w+V(x)w = \varepsilon^{-\theta}W(x)(I_\theta*(W|w|^p))|w|^{p-2}w,\quad x\in\mathbb{R}^N, ~~~~~~~~~~~~(*)\end{equation} $\end{document} where \begin{document}$ \varepsilon>0, N>2, I_\theta $\end{document} is the Riesz potential with order \begin{document}$ \theta\in(0,N), p\in\big[2,\frac{N+\theta}{N-2}\big), \min V>0 $\end{document} and \begin{document}$ \inf W>0 $\end{document} . Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for \begin{document}$ (\ast) $\end{document} . The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set.