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Multiplicity and concentration of dual solutions for a Helmholtz system

Ruowen Qiu, Fei Yuan, Fukun Zhao

TL;DR

This paper studies the nonlinear Helmholtz system of Hamiltonian type in $\mathbb{R}^N$ ($N\ge3$) given by $-\Delta u-k^2u=P(x)|v|^{p-2}v$ and $-\Delta v-k^2v=Q(x)|u|^{q-2}u$, establishing the existence of dual ground states via a novel dual variational framework built on the Birman-Schwinger operator. It analyzes the concentration behavior as the frequency parameter $k\to\infty$ through a rescaling argument and a limit problem, showing convergence to a ground state of the constant-coefficient system and localization near the maxima of $P$ and $Q$. Furthermore, the authors prove multiplicity results: for large $k$, at least $\mathrm{cat}_{M_\delta}(M)$ pairs of dual solutions exist, where $M$ is tied to the joint maxima of $P$ and $Q$, and $M_\delta$ is a neighborhood controlling concentration. The work combines dual variational methods, nonlocal operator theory, and Lusternik-Schnirelman category to connect spectral properties, concentration phenomena, and topological complexity of the weight sets, extending the theory of Helmholtz systems of Hamiltonian type.

Abstract

In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -Δv-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $N\geq3$, $P,Q: \mathbb{R}^N\rightarrow \mathbb{R}$ are two positive continuous functions, the exponents $p,q>2$ satisfy $\frac{1}{p}+\frac{1}{q}>\frac{N-2}{N}$. First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as $k\rightarrow\infty$, where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions $P$ and $Q$.

Multiplicity and concentration of dual solutions for a Helmholtz system

TL;DR

This paper studies the nonlinear Helmholtz system of Hamiltonian type in () given by and , establishing the existence of dual ground states via a novel dual variational framework built on the Birman-Schwinger operator. It analyzes the concentration behavior as the frequency parameter through a rescaling argument and a limit problem, showing convergence to a ground state of the constant-coefficient system and localization near the maxima of and . Furthermore, the authors prove multiplicity results: for large , at least pairs of dual solutions exist, where is tied to the joint maxima of and , and is a neighborhood controlling concentration. The work combines dual variational methods, nonlocal operator theory, and Lusternik-Schnirelman category to connect spectral properties, concentration phenomena, and topological complexity of the weight sets, extending the theory of Helmholtz systems of Hamiltonian type.

Abstract

In this paper, we are concerned with the nonlinear Helmholtz system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu-k^2 u=P(x)|v|^{p-2}v,\quad \text{in}\ \mathbb{R}^N, \\ -Δv-k^2v=Q(x)|u|^{q-2}u,\quad \text{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where , are two positive continuous functions, the exponents satisfy . First, we obtained the existence of a ground state solution via a dual variational method. Moreover, the concentration behavior of such dual ground state solutions is established as , where a rescaling technique and the generalized Birman-Schwinger operator are involved. In addition, we also investigated the relation between the number of solutions and the topology of the set of the global maxima of the functions and .
Paper Structure (6 sections, 14 theorems, 120 equations)

This paper contains 6 sections, 14 theorems, 120 equations.

Table of Contents

  1. Introduction and main result
  2. The variational formulation and preliminary results
  3. Dual variational formulation
  4. Preliminaries
  5. Palais-Smale condition and representation Lemma
  6. The proof of Theorem 1.1 and Theorem 1.2

Key Result

Theorem 1.1

Let $N \geq 3$ and $p$, $q>\frac{2 N}{N-1}$ with $\frac{N-2}{N}<\frac{1}{p}+\frac{1}{q}<\frac{N-1}{N+1}$. If $(PQ_1)$$-$$(PQ_3)$ hold, then there is $k_0>0$ such that for all $k>k_0$ the system eq1.1 admits a pair of dual ground state. In addition, let $\{k_n\}\subset(k_0,\infty)$ with $k_n\rightarr Then there exists a pair of $(x_0,x_0)\in M$, a dual ground state $(u_0,v_0)$ of and a sequence $\

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 5 more