Localized nodal solutions of higher topological type for nonlinear Schrödinger–Poisson system
Section snippets
Introduction and main results
Consider following nonlinear Schrödinger–Poisson system where and are small positive parameters. Principal aim of this paper is to construct pairs of sign-changing solutions of system (1.1) for any positive integer , and study the concentration phenomena of these semiclassical states as . The potential satisfies following assumption:
and there exist such that
- ()
there exists a bounded domain with
Preliminaries
In this section, we first recall some properties of function , then introduce a new minimax theorem in the presence of invariant sets of negative gradient flow from [31].
To investigate (1.1), we will focus our attention on the following equivalent system by making the change of variable , and the corresponding energy functional is where is the unique solution of
The Palais–Smale condition for penalized functional
To deal with the compactness issue of Palais–Smale sequence for functional , we apply the penalization method employed by Byeon and Wang [11].
For any and any set , we set Let be a cut-off function so that and for every , if , if and if . Define For small, one can verify that is a function. For , we define where .
Existence of multiple sign-changing critical points of
In this section, we will apply the abstract minimax Theorem 2.1 to obtain multiple sign-changing critical points of the penalized functional .
Define For , set where for and . It is easy to see that . Then we take in Definition 2.1 and Theorem 2.1. Clearly, is an open and symmetric subset of and
Proof of main results
In this section, we will prove our main theorems by careful observation and analysis on the asymptotic behaviors of critical point for the penalized functional , and detailed estimates on the local Pohozaev identity associated with .
Lemma 5.1 For any and , there exist a positive constant depending only on and and a positive constant independent of and such that
Proof By a standard argument, we can prove this lemma, see, for instance, [15, Lemma
Acknowledgments
This work was supported by the NSFC (Nos. 11801574, 11971485, 11601145, 11701173), Natural Science Foundation of Hunan Province (No. 2019JJ50788), Central South University Innovation-Driven Project for Young Scholars (No. 2019CX022), and Excellent youth project of Education Department of Hunan Province (18B342), PR China.
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