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Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

Jiao Luo, Zhipeng Yang

Abstract

We study a logarithmic fractional Schrödinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+φu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αφ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here \(α\in\bigl(\frac34,1\bigr)\), \(4<p<2_α^{*}=\frac{6}{3-2α}\), and \(V\) satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a \(C^{1}\) variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed \(δ>0\) and all sufficiently small \(\varepsilon>0\), the system admits at least \(\operatorname{cat}_{M_δ}(M)\) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of \(V\) as \(\varepsilon\to0\).

Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

Abstract

We study a logarithmic fractional Schrödinger--Poisson system in : \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+φu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αφ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here \(α\in\bigl(\frac34,1\bigr)\), , and satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed and all sufficiently small , the system admits at least \(\operatorname{cat}_{M_δ}(M)\) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of as .

Paper Structure

This paper contains 17 sections, 33 theorems, 555 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Variational Framework
  3. Orlicz spaces and fractional Sobolev spaces
  4. The logarithmic decomposition and the working space
  5. Nehari manifold and ground states
  6. Geometry of the Nehari manifold
  7. The mountain-pass level and its characterization
  8. Compactness at the mountain-pass level
  9. Existence of a ground state solution
  10. Autonomous problem and concentration tools
  11. The autonomous family and ground states
  12. Translation compactness of low-energy Nehari sequences
  13. Uniform boundedness and decay estimates
  14. Multiplicity and proof of the main theorem
  15. Low-energy compactness on the Nehari manifold
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume that $\alpha\in\bigl(\frac{3}{4},1\bigr)$, $4<p<2_\alpha^*$, and $(V)$ holds. Then, for every fixed $\delta>0$, there exists $\varepsilon_1=\varepsilon_1(\delta)>0$ such that, for every $\varepsilon\in(0,\varepsilon_1)$, problem eq1.1 has at least $\operatorname{cat}_{M_\delta}(M)$ distinct p $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (65)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 55 more