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Normalized solutions for a fractional Schrödinger-Poisson system with critical growth

Xiaoming He, Yuxi Meng, Marco Squassina

Abstract

In this paper, we study the fractional critical Schrödinger-Poisson system \[\begin{cases} (-Δ)^su +λφu= αu+μ|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-Δ)^tφ=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $λ,μ>0$ parameters and $α\in{\mathbb R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.

Normalized solutions for a fractional Schrödinger-Poisson system with critical growth

Abstract

In this paper, we study the fractional critical Schrödinger-Poisson system having prescribed mass where satisfies and parameters and is an undetermined parameter. Under the -subcritical perturbation , we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the -supercritical perturbation , by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.
Paper Structure (8 sections, 28 theorems, 302 equations)

This paper contains 8 sections, 28 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Remarks on the proofs
  4. Paper outline
  5. Preliminary stuff
  6. Proof of Theorem \ref{['Theorem 2.1']}
  7. Proof of Theorem \ref{['Theorem 2.2']}
  8. Proof of Theorem \ref{['Theorem 2.3']}

Key Result

Theorem 2.1

Let $\mu,\lambda, a>0$, and $q\in (2,2+\frac{4s}{3})$. Then, for a given $k\in \mathbb{N}$, there exists $\beta>0$ independent of $k$ and $\mu^*_k>0$ large, such that problem e1.5-e1.6 possesses at least $k$ couples $(u_j,\alpha_j)\in H^s(\mathbb R^3)\times \mathbb R$ of weak solutions for $\mu>\mu_ with $\int_{\mathbb R^3}|u_j|^2dx=a^2$, $\alpha_j<0$ for all $j=1,\cdots,k$.

Theorems & Definitions (44)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 34 more